Exponentials and Logarithms: Rules, Equation & Examples

Exponentials and logarithms are related mathematical concepts – in fact, they are inverse functions of each other. Having an understanding of how to use both exponential and logarithms will help you to better understand and manipulate more complicated functions.

What are exponentials?

An exponential is a type of function written in the form $y = a^{x}$ . Here, a is a positive constant named the base, whilst x is the variable, named the exponent or indices.
It is common to see exponential functions with a base of e, e.g. $y = e^{x}$ . This is short for Euler's number (2.71828 ...).

Use the e button on the calculator rather than the specific number itself.

Exponentials tell you what the answer would be if you repeatedly multiplied the same constant by itself. For example, $3^{4} = 3 \times 3 \times 3 \times 3$
Graphically, exponentials will all have a similar shape. Here is $y = 6^{x}$ :

Exponential rules

Exponential rules are also referred to as power or indices rules. Here's a recap of exponential rules:

Rule	Example
$x^{0}$ = 1	$6^{0} = 1$
$x^{- a} = \frac{1}{x^{a}}$	$5^{- 2} = \frac{1}{5^{2}} = \frac{1}{25}$
$x^{a} \times x^{b} = x^{a + b}$	$2^{2} \times 2^{3} = 2^{2 + 3} = 2^{5} = 32$
$\frac{x^{a}}{x^{b}} = x^{a - b}$	$\frac{8^{5}}{8^{3}} = 8^{2} = 64$
${(x^{m})}^{n} = x^{m \times n}$	${(3^{2})}^{3} = 3^{2 \times 3} = 3^{6} = 729$

To use exponential rules the bases need to be the same.

What are logarithms?

Logarithms, or logs for short, are the inverse of exponential functions and are used when we do not know what the exponent (power) is.

Dissecting logarithms

Logarithms are written in the form $L o g_{a} b$ to answer the question $a^{x} = b$ to find x .

a is the base and is the constant being raised to a power. The most common base is 10 and as a result, where there is no base visible in the question (eg log (15)), the base is 10.
b is the answer to the exponential $(a^{x} = b)$
x is the exponent

A logarithm with a base of e has its own logarithm In (x): called the natural logarithm.

Using logarithms to find the exponent

We can therefore use logarithms to solve exponentials with a missing exponent.

Identify the base, answer of the exponential and exponent.
Rewrite as a logarithm in the form $L o g_{b a s e} (a n s w e r t o e x p o n t e n t i a l) = e x p o n e n t$
Rearrange if necessary
Calculate using a calculator

Solve $5^{x} = 625$

Base: 5, Answer of exponential: 625, exponent: x
x = $L o g_{5} (625)$
x = $L o g_{5} (625) = 4$

To work out more complicated exponents, you do the same method.

Solve $2^{3 x + 1} = 50$

Base: 2, Answer of exponential: 50, exponent: 3x + 1
$L o g_{2} (50) = 3 x + 1$
$L o g_{2} (50) - 1 = 3 x$
$\frac{L o g_{2} (50) - 1}{3} = x = 1.55 (3 s . f)$

If you are asked to give your answer in its exact form, leave it in its log form.

Overview of logarithm rules

Logarithm rules can be used to simplify and solve logarithms. As with exponential functionals, to use logarithm rules all the bases need to be the same.

The basic rules:

$L o g (x) + L o g (y) = L o g (x y)$
$L o g (x) - L o g (y) = L o g (\frac{x}{y})$
$L o g (x^{a}) = a L o g (x)$

The more complicated ones:

$L o g (\frac{1}{x}) = L o g (x^{- 1}) = - L o g (x)$
$L o g_{a} (a) = 1$
$L o g_{a} (1) = 0$

Exponentials and Logarithms - Key takeaways

Exponentials and logarithms are inverse functions of each other. They use the same information but solve for different variables. Exponential (indices) functions are used to solve when a constant is raised to an exponent (power), whilst a logarithm solves to find the exponent.
Both exponentials and logarithms have their own rules that you need to use.

Frequently Asked Questions about Exponentials and Logarithms

No. They are similar as they both use the same information (bases, exponents, answer of base to the power of exponent). However, they solve to give you different information.

Logarithms (logs) and exponentials are inverse functions; therefore, exponentials are the opposite of a log, and logs are the opposite of exponentials.

Both exponentials and logarithms use the same information but differ in what they find. An exponential is used to find the value of the base raised to an exponent, whilst a logarithm is used to find the exponent (power).

As they are inverse functions, switching between logarithms and exponentials does not need mathematical manipulation. Simply label each part of the function and rearrange it into the other functions form. For instance, Log Base (answer of exponential) = exponent goes to Base Exponent = Answer of exponential.

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