Logarithms, or logs for short, are essentially powers and are useful when a power is unknown.

When you come across logs, you will usually see the word ‘log’ followed by a small subscript then a number in brackets:

$\log_a(b)=c.$

The subscript is known as the base and the number in brackets (although sometimes the brackets are left out) is the exponent.

It can help to understand logs by making a habit of, when reading log expressions, saying ‘the power of’ instead of the word ‘log’. It follows that the above reads as ‘the power of a to get a result of b is c’. For example, in $\log_2(8)$ the base is 2 and since it can be read as the power of 2 to give 8, the value of $\log_2(8)$ is 3. $\log_a(x)$ is considered to be the inverse of $a^x$see Logarithmic Graphs.

## Non-Calculator Logs Examples

The equation $\log_a(b)=c$ can be used interchangeably with the equation

$a^c=b$.

For example, $\log_5(25)=x$ can be changed to $5^x=25$ and so x is 2.

Some other examples include:
$\begin{array}{l}\log_3(9)=2\\\log_4(64)=3\\\log_2(1/8)=-3 \text{ since }2^{-3}=\frac{1}{2^3}=\frac{1}{8}\\\log_{123}(1)=0\end{array}$ since anything to the power of 0 is 1.

## Calculator Logs Examples

There is a button on your calculator that can help you with powers that are not calculable in your head. The button has the word log followed by a two boxes; insert your base into the small lower box and the number that you see in brackets in the second. Use it to verify the following:

$\begin{array}{l}\log_{9}(5)=0.732 \text{ to 3 decimal places}\\\log_{8}(9)=1.06 \text{ to 2 decimal places}\\\log_{12}(150)=2.016 \text{ to 3 decimal places}\\\log_{2}(0.7)=-0.515 \text{ to 3 decimal places}\\\log_{3}(-4)= \text{ not possible}^* \end{array}$

* Note that logging negative numbers with a positive base is not possible. This is because, there is no amount of times you can multiply a positive number by itself to get any negative number. Your calculator will verify this with a MATH ERROR.

Note that $\log_a(a)=1$ since $a^1=a$. Note also that log to the base e is the natural logarithm and has the special notation ln. For example, $\log_e(8)=\ln(8)$. Note that logs used to be calculated on a slide rule – click here to find out more. Also check out log graphs.

## The Logs Rules

Arguably the most important identity when using logs is the following:

$\log_a(b)=c\hspace{10pt}\Longleftrightarrow\hspace{10pt}a^c=b$

This is because log equations (see below) can be solved once in this format.

The following rules follow from the Laws of Indices and should be used to convert log expressions into a single log.

$\begin{array}{c}\log_a(x)+\log_a(y)=\log_a(xy) \hspace{20pt}(1)\\\log_a(x)-\log_a(y)=\log_a(x/y)\hspace{20pt}(2)\\\log_a(x^n)=n\log_a(x)\hspace{20pt}(3)\end{array}$

You can also change the base of the log using the following formula:

$\log_a(x)=\frac{\log_b(x)}{\log_b(a)}$

See Example 4 below to see an example of using a change of base.

## Example 1

Write the following as a single logarithm:

$3\log_5(2)+2\log_5(4)-\log_5(6)$

## Example 2

Expand the following in terms of logs of x, y and z:

$\log_a\left(\frac{x^2\sqrt{y}}{z^5}\right)$

## Solving Equations using Logs

The above log identities and rules can be used to solve log equations. Of course, if an equation is in the form $a^c=b$, logs can be applied to solve the equation. Additionally, if the equation involves an expression that is not yet in the format $\log_a{b}$, use the log rules to get it in this format.

### Example 1

Find the exact solution of i) $4e^{2x-7}=20$ and ii) $\ln(3y-5)=6$.

### Example 2

Solve the equation $5^{x}+9\times 5^{-x}=6$.

### Example 3

Solve $2^xe^{3x}=e^5$.

### Example 4

Solve the equation   $\log_2(x)=\log_x(5)$.

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