# Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions

## Inverse Functions

An inverse function is a function that undoes another function.### Learning Objectives

Discuss what it means to be an inverse function### Key Takeaways

#### Key Points

- If an input [latex]x[/latex] into the function [latex]f[/latex] produces an output [latex]y[/latex], then putting [latex]y[/latex] into the inverse function [latex]g[/latex] produces the output [latex]x[/latex], and vice versa (i.e., [latex]f(x)=y[/latex], and [latex]g(y)=x[/latex]).
- A function [latex]f[/latex] that has an inverse is called invertible; the inverse function is then uniquely determined by [latex]f[/latex] and is denoted by [latex]f^{-1}[/latex].
- If [latex]f[/latex] is invertible, the function [latex]g[/latex] is unique; in other words, there is exactly one function [latex]g[/latex] satisfying this property (no more, no fewer).

#### Key Terms

**inverse**: a function that undoes another function**function**: a relation in which each element of the domain is associated with exactly one element of the co-domain

### Example

Let's take the function [latex]y=x^2+2[/latex]. To find the inverse of this function, undo each of the operations on the [latex]x[/latex] side of the equation one at a time. We start with the [latex]+2[/latex] operation. Notice that we start in the opposite order of the normal order of operations when we undo operations. The opposite of [latex]+2[/latex] is [latex]-2[/latex]. We are left with [latex]x^2[/latex]. To undo use the square root operation. Thus, the inverse of [latex]x^2+2[/latex] is [latex]\sqrt{x-2}[/latex]. We can check to see if this inverse "undoes" the original function by plugging that function in for [latex]x[/latex]: [latex-display]\sqrt{\left(x^2+2\right)-2}=\sqrt{x^2}=x[/latex-display]## Derivatives of Exponential Functions

The derivative of the exponential function is equal to the value of the function.### Learning Objectives

Solve for the derivatives of exponential functions### Key Takeaways

#### Key Points

- [latex]e^x[/latex] is its own derivative: [latex]\frac{d}{dx}e^{x} = e^{x}[/latex].
- If a variable 's growth or decay rate is proportional to its size, then the variable can be written as a constant times an exponential function of time.
- For any differentiable function [latex]f(x)[/latex], [latex]\frac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}[/latex].

#### Key Terms

**exponential**: any function that has an exponent as an independent variable**tangent**: a straight line touching a curve at a single point without crossing it there**e**: the base of the natural logarithm, [latex]2.718281828459045\dots[/latex]

- The slope of the graph at any point is the height of the function at that point.
- The rate of increase of the function at [latex]x[/latex] is equal to the value of the function at [latex]x[/latex].
- The function solves the differential equation [latex]y' = y [/latex].
- [latex]e^x[/latex] is a fixed point of derivative as a functional.

## Logarithmic Functions

The logarithm of a number is the exponent by which another fixed value must be raised to produce that number.### Learning Objectives

Demonstrate that logarithmic functions are the inverses of exponential functions### Key Takeaways

#### Key Points

- The idea of logarithms is to reverse the operation of exponentiation, that is raising a number to a power.
- A naive way of defining the logarithm of a number [latex]x[/latex] with respect to base [latex]b[/latex] is the exponent by which [latex]b[/latex] must be raised to yield [latex]x[/latex].
- To define the logarithm, the base [latex]b[/latex] must be a positive real number not equal to [latex]1[/latex] and [latex]x[/latex] must be a positive number.

#### Key Terms

**binary**: the bijective base-2 numeral system, which uses only the digits 0 and 1**exponent**: the power to which a number, symbol or expression is to be raised: for example, the [latex]3[/latex] in [latex]x^3[/latex].

^{3}. More generally, if [latex]x = b^y[/latex], then [latex]y[/latex] is the logarithm of [latex]x[/latex] to base [latex]b[/latex], and is written [latex]y=\log_b(x)[/latex], so [latex]\log_{10}(1000)=3[/latex] log

_{10}(1000) = 3. The logarithm to base [latex]b = 10[/latex] is called the common logarithm and has many applications in science and engineering. The natural logarithm has the constant [latex]e[/latex] ([latex]\approx 2.718[/latex]) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base [latex]b = 2[/latex] and is prominent in computer science. The idea of logarithms is to reverse the operation of exponentiation, that is raising a number to a power. For example, the third power (or cube) of 2 is 8, because 8 is the product of three factors of 2: [latex]2^{3} = 2 \times 2\times 2 = 8[/latex]. It follows that the logarithm of 8 with respect to base 2 is 3, so log

_{2}8 = 3. A naive way of defining the logarithm of a number [latex]x[/latex] with respect to base b is the exponent by which b must be raised to yield x. In other words, the logarithm of [latex]x[/latex] to base b is the solution y to the equation: [latex]b^{y} = x[/latex]. This definition assumes that we know exactly what we mean by 'raising a real positive number to a real power'. Raising to integer powers is easy. It is clear that two raised to the third is eight, because 2 multiplied by itself 3 times is 8, so the logarithm of eight with respect to base two will be 3. However, the definition also assumes that we know how to raise numbers to non-integer powers. What would be the logarithm of ten? The definition tells us that the binary logarithm of ten is 3.3219 because two raised to the 3.3219th power is ten. So, the definition only makes sense if we know how to multiply 2 by itself 3.3219 times. For the definition to work, it must be understood that ' raising two to the 0.3219 power' means 'raising the 10000th root of 2 to the 3219th power'. The ten-thousandth root of 2 is 1.0000693171 and this number raised to the 3219th power is 1.2500, therefore ' 2 multiplied by itself 3.3219 times' will be 2 x 2 x 2 x 1.2500 namely 10. Making this proviso, if the base b is any positive number except 1, and the number [latex]x[/latex] is greater than zero, there is always a real number y that solves the equation: [latex]b^{y} = x[/latex] so the logarithm is well defined. The logarithm is denoted "log

_{b}(x)". In the equation y = log

_{b}(x), the value y is the answer to the question "To what power must b be raised, in order to yield x?". To define the logarithm, the base b must be a positive real number not equal to 1 and x must be a positive number.

## Derivatives of Logarithmic Functions

The general form of the derivative of a logarithmic function is [latex]\frac{d}{dx}\log_{b}(x) = \frac{1}{xln(b)}[/latex].### Learning Objectives

Solve for the derivative of a logarithmic function### Key Takeaways

#### Key Points

- The derivative of natural logarithmic function is [latex]\frac{d}{dx}\ln(x) = \frac{1}{x}[/latex].
- The general form of the derivative of a logarithmic function can be derived from the derivative of a natural logarithmic function.
- Properties of the logarithm can be used to to differentiate more difficult functions, such as products with many terms, quotients of composed functions, or functions with variable or function exponents.

#### Key Terms

**logarithm**: the exponent by which another fixed value, the base, must be raised to produce that number**e**: the base of the natural logarithm, [latex]2.718281828459045\dots[/latex]

- [latex]\log \left(\dfrac{a}{b}\right) = \log (a) - \log (b)[/latex]
- [latex]\log(a^{n}) = n \log(a)[/latex]
- [latex]\log(a) + \log (b) = \log(ab)[/latex]

## The Natural Logarithmic Function: Differentiation and Integration

Differentiation and integration of natural logarithms is based on the property [latex]\frac{d}{dx}\ln(x) = \frac{1}{x}[/latex].### Learning Objectives

Practice integrating and differentiating the natural logarithmic function### Key Takeaways

#### Key Points

- The natural logarithm allows simple integration of functions of the form [latex]g(x) = \frac{ f '(x)}{f(x)}[/latex].
- The natural logarithm can be integrated using integration by parts: [latex]\int\ln(x)dx=x \ln(x)−x+C[/latex].
- The derivative of the natural logarithm leads to the Taylor series for [latex]\ln(1 + x)[/latex] around [latex]0[/latex]: [latex]\ln(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \cdots[/latex] for [latex]\left | x \right | \leq 1[/latex] (unless [latex]x = -1[/latex]).

#### Key Terms

**transcendental**: of or relating to a number that is not the root of any polynomial that has positive degree and rational coefficients**irrational**: of a real number, that cannot be written as the ratio of two integers

## The Natural Exponential Function: Differentiation and Integration

The derivative of the exponential function [latex]\frac{d}{dx}a^x = \ln(a)a^{x}[/latex].### Learning Objectives

Practice integrating and differentiating the natural exponential function### Key Takeaways

#### Key Points

- The formula for differentiation of exponential function [latex]a^{x}[/latex] can be derived from a specific case of natural exponential function [latex]e^{x}[/latex].
- The derivative of the natural exponential function [latex]e^{x}[/latex] is expressed as [latex]\frac{d}{dx}e^{x} =e^{x}[/latex].
- The integral of the natural exponential function [latex]e^{x}[/latex] is [latex]\int e^{x}dx = e^{x} + C[/latex].

#### Key Terms

**differentiation**: the process of determining the derived function of a function**e**: the base of the natural logarithm, [latex]2.718281828459045\dots[/latex]

## Exponential Growth and Decay

Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value.### Learning Objectives

Apply the exponential growth and decay formulas to real world examples### Key Takeaways

#### Key Points

- The formula for exponential growth of a variable [latex]x[/latex] at the (positive or negative) growth rate [latex]r[/latex], as time [latex]t[/latex] goes on in discrete intervals (that is, at integer times [latex]0, 1, 2, 3, \cdots[/latex]), is: [latex]x_{t} = x_{0}(1 + r^{t})[/latex] where [latex]x_0[/latex] is the value of [latex]x[/latex] at time [latex]0[/latex].
- Exponential decay occurs in the same way as exponential growth, providing the growth rate is negative.
- In the long run, exponential growth of any kind will overtake linear growth of any kind as well as any polynomial growth.

#### Key Terms

**exponential**: any function that has an exponent as an independent variable**linear**: having the form of a line; straight**polynomial**: an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power

## Inverse Trigonometric Functions: Differentiation and Integration

It is useful to know the derivatives and antiderivatives of the inverse trigonometric functions.### Learning Objectives

Practice integrating and differentiating inverse trigonometric functions### Key Takeaways

#### Key Points

- The inverse trigonometric functions "undo" the trigonometric functions [latex]\sin[/latex], [latex]\cos[/latex], and [latex]\tan[/latex].
- The inverse trigonometric functions are [latex]\arcsin[/latex], [latex]\arccos[/latex], and [latex]\arctan[/latex].
- Memorizing their derivatives and antiderivatives can be useful.

#### Key Terms

**trigonometric**: relating to the functions used in trigonometry: [latex]\sin[/latex], [latex]\cos[/latex], [latex]\tan[/latex], [latex]\csc[/latex], [latex]\cot[/latex], [latex]\sec[/latex]

## Hyperbolic Functions

[latex]\sinh[/latex] and [latex]\cosh[/latex] are basic hyperbolic functions; [latex]\sinh[/latex] is defined as the following: [latex]\sinh (x) = \frac{e^x - e^{-x}}{2}[/latex].### Learning Objectives

Discuss the basic properties of hyperbolic functions### Key Takeaways

#### Key Points

- The basic hyperbolic functions are the hyperbolic sine "[latex]\sinh[/latex]," and the hyperbolic cosine "[latex]\cosh[/latex]," from which are derived the hyperbolic tangent "[latex]\tanh[/latex]," and so on, corresponding to the derived trigonometric functions.
- The inverse hyperbolic functions are the area hyperbolic sine "[latex]\text{arsinh}[/latex]" (also called "[latex]\text{asinh}[/latex]" or sometimes "[latex]\text{arcsinh}[/latex]") and so on.
- The hyperbolic functions take real values for a real argument called a hyperbolic angle. The size of a hyperbolic angle is the area of its hyperbolic sector.

#### Key Terms

**meromorphic**: relating to or being a function of a complex variable that is analytic everywhere in a region except for singularities at each of which infinity is the limit and each of which is contained in a neighborhood where the function is analytic except for the singular point itself**inverse**: a function that undoes another function

### Hyperbolic sine

[latex-display]\sinh (x) = \dfrac{e^x - e^{-x}}{2}[/latex-display]### Hyperbolic cosine

[latex-display]\cosh (x) =\dfrac{e^x + e^{-x}}{2}[/latex-display]### Hyperbolic tangent

[latex-display]\tanh (x) = \dfrac{\sinh(x)}{\cosh (x)} = \dfrac{1 - e^{-2x}}{1 + e^{-2x}}[/latex-display]### Hyperbolic cotangent

[latex-display]\coth (x) = \dfrac{\cosh (x)}{\sinh (x)} = \dfrac{1 + e^{-2x}}{1 - e^{-2x}}[/latex-display]### Hyperbolic secant

[latex-display]\text{sech} (x) = (\cosh (x))^{-1}= \dfrac{2}{e^x + e^{-x}}[/latex-display]### Hyperbolic cosecant

[latex-display]\text{csch} (x)= (\sinh (x))^{-1}= \dfrac{2}{e^x - e^{-x}}[/latex-display]## Indeterminate Forms and L'Hôpital's Rule

Indeterminate forms like [latex]\frac{0}{0}[/latex] have no definite value; however, when a limit is indeterminate, l'Hôpital's rule can often be used to evaluate it.### Learning Objectives

Use L'Hopital's Rule to evaluate limits involving indeterminate forms### Key Takeaways

#### Key Points

- Indeterminate forms include [latex]0^0[/latex], [latex]\frac{0}{0}[/latex], [latex]1^\infty[/latex], [latex]\infty - \infty[/latex], [latex]\frac{\infty}{\infty}[/latex], [latex]0 \times \infty[/latex], and [latex]\infty^0[/latex]
- Indeterminate forms often arise when you are asked to take the limit of a function. For example: [latex]\lim_{x\to 0}\frac{x}{x}[/latex] is indeterminate, giving [latex]\frac{0}{0}[/latex].
- L'Hôpital's rule: For [latex]f[/latex] and [latex]g[/latex] which are differentiable, if [latex]\lim_{x\to c}f(x)=\lim_{x \to c}g(x) = 0[/latex] or [latex]\pm \infty[/latex] and [latex]\lim_{x\to c}\frac{f'(x)}{g'(x)}[/latex] exists, and [latex]g'(x) \neq 0[/latex] for all [latex]x[/latex] in the interval containing [latex]c[/latex], then [latex]\lim_{x \to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}[/latex].

#### Key Terms

**limit**: a value to which a sequence or function converges**differentiable**: a function that has a defined derivative (slope) at each point**indeterminate**: not accurately determined or determinable

### L'Hôpital's Rule

In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit. In its simplest form, l'Hôpital's rule states that for functions [latex]f[/latex] and [latex]g[/latex] which are differentiable, if [latex-display]\displaystyle{\lim_{x\to c}f(x)=\lim_{x \to c}g(x) = 0 \text{ or } \pm \infty}[/latex-display] and [latex]\lim_{x\to c}\frac{f'(x)}{g'(x)}[/latex] exists, and [latex]g'(x) \neq 0[/latex] for all [latex]x[/latex] in the interval containing [latex]c[/latex], then: [latex-display]\displaystyle{\lim_{x \to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}}[/latex-display]## Bases Other than e and their Applications

Among all choices for the base [latex]b[/latex], particularly common values for logarithms are [latex]e[/latex], [latex]2[/latex], and [latex]10[/latex].### Learning Objectives

Distinguish between the different applications for logarithms in various bases### Key Takeaways

#### Key Points

- The major advantage of common logarithms (logarithms to base ten) is that they are easy to use for manual calculations in the decimal number system.
- The binary logarithm is often used in computer science and information theory because it is closely connected to the binary numeral system.
- Common logarithm is frequently written as "[latex]\log(x)[/latex]"; binary logarithm is frequently written "[latex]\text{ld}\, n[/latex]" or "[latex]\lg n[/latex]".

#### Key Terms

**logarithm**: the exponent by which another fixed value, the base, must be raised to produce that number

## Licenses & Attributions

### CC licensed content, Shared previously

- Curation and Revision.
**Provided by:**Boundless.com**License:**CC BY-SA: Attribution-ShareAlike.

### CC licensed content, Specific attribution

- Inverse function.
**Provided by:**Wikipedia**Located at:**https://en.wikipedia.org/wiki/Inverse_function.**License:**CC BY-SA: Attribution-ShareAlike. - inverse.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - function.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**Located at:**https://upload.wikimedia.org/wikipedia/commons/9/9d/Inverse_Functions_Domain_and_Range.png.**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**Located at:**https://upload.wikimedia.org/wikipedia/commons/c/c8/Inverse_Function.png.**License:**CC BY-SA: Attribution-ShareAlike. - Exponential function.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - exponential.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - tangent.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - e.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Logarithmic function.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Logarithmic function.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - exponent.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - binary.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary_logarithm_plot_with_ticks.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Calculus/Derivatives of Exponential and Logarithm Functions.
**Provided by:**Wikibooks**License:**CC BY-SA: Attribution-ShareAlike. - Logarithmic functions.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - logarithm.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - e.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary_logarithm_plot_with_ticks.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Natural logarithm.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Natural logarithm.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Natural logarithm.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - transcendental.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - irrational.
**Provided by:**Wiktionary**Located at:**https://en.wiktionary.org/wiki/irrational.**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary_logarithm_plot_with_ticks.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - LogTay.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Calculus/Derivatives of Exponential and Logarithm Functions.
**Provided by:**Wikibooks**License:**CC BY-SA: Attribution-ShareAlike. - Boundless.
**Provided by:**Boundless Learning**License:**CC BY-SA: Attribution-ShareAlike. - e.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary_logarithm_plot_with_ticks.png.
**Provided by:**Wikipedia**Located at:**https://en.wikipedia.org/wiki/Logarithm.**License:**CC BY-SA: Attribution-ShareAlike. - LogTay.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exp.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exponential growth.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exponential growth.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - polynomial.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - linear.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - exponential.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary_logarithm_plot_with_ticks.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - LogTay.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exp.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exponential.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - List of integrals of inverse trigonometric functions.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Inverse trigonometric functions.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Differentiation of trigonometric functions.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - trigonometric.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary_logarithm_plot_with_ticks.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - LogTay.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exp.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exponential.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arctangent_Arccotangent.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arcsecant_Arccosecant.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arcsine_Arccosine.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Hyperbolic function.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - meromorphic.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - inverse.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary_logarithm_plot_with_ticks.png.
**Provided by:**Wikipedia**Located at:**https://en.wikipedia.org/wiki/Logarithm.**License:**CC BY-SA: Attribution-ShareAlike. - LogTay.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exp.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exponential.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arctangent_Arccotangent.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arcsecant_Arccosecant.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arcsine_Arccosine.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - L'Hu00f4pital's rule.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Indeterminate form.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - limit.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - indeterminate.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. - differentiable.
**Provided by:**Wiktionary**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary_logarithm_plot_with_ticks.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - LogTay.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exp.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exponential.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arctangent_Arccotangent.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arcsecant_Arccosecant.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arcsine_Arccosine.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary logarithm.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Logarithm.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - logarithm.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. -
**Provided by:**Wikimedia**License:**CC BY-SA: Attribution-ShareAlike. - Binary_logarithm_plot_with_ticks.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - LogTay.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exp.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Exponential.png.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arctangent_Arccotangent.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arcsecant_Arccosecant.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike. - Arcsine_Arccosine.svg.
**Provided by:**Wikipedia**License:**CC BY-SA: Attribution-ShareAlike.