mathematics > csec > additional mathematics > algebra and functions > logarithms

# Logarithms

The log of a number is the **power** to which the base must raised in order to produce that number.

Given the *exponential form*:
$$a^x=b$$
$a$ is the base, $x$ is the exponent and $b$ is the given number. We can rewrite this in *log form*:
$$log_{a}b=x$$
We read this as **log base a of b is x**. Notice how $x$ is the power in the exponential form. *$x$ is the log* (the log is the power).

We know that $3^2=9$ therefore in log form this will be $$log_{3}9=2$$ 3 is the base, 9 is the given number and 2 is the power (therefore log base 3 of 9 is 2 because the log is the power and 2 is the power).

#### Challenge

Given the exponential form $4^2=16$, what is the log form? (Tap/click an answer below)

Given the logarithmic form $log_{3}81=4$, what is the exponential form? (Tap/click an answer below)

Find the value of $x$ in $log_{12}x=2$ (Tap/click an answer below)

The logarithm is (Tap/click an answer below)

If $a^0=1$ then $log_{a}1=?$ (Tap/click an answer below)

## Laws of logs

There are three (3) basic laws of logs.

### The sum of two logs

$log_{a}b+log_{a}c=log_{a}bc$

### The difference of two logs

$log_{a}b-log_{a}c=log_{a}\frac{b}{c}$

### The scalar multiple of a log

$nlog_{a}b$=$log_{a}b^n$