Logarithmic Functions

Lesson

We've learnt about indices and ways we can manipulate them in maths, and now it's time to let you in on the secret that there's actually a much fancier name for them - exponentials! Yes you've probably heard of this word before when talking about something growing exponentially to mean it increasing very very fast, just like indices do to a base as well.

In maths we have pairs of inverse functions, or opposite operations, eg. addition & subtraction, multiplication & division. The inverse function of an exponential is called a logarithm, and it works like this:

Logarithmic form

If we have an exponential of the form:

$b^x=a$`b``x`=`a`

Then we can rewrite it as the logarithm:

$\log_ba=x$`l``o``g``b``a`=`x`

and $b$`b` is called the base, just like in exponentials

In other words we use logarithms when we are interested in finding out the index needed ($x$`x`) to raise a certain base ($b$`b`) to a certain number ($a$`a`).

Careful!

An important part to remember is that you can only take the logarithm of a positive number with a positive base that's not one:

$\log_ba$`l``o``g``b``a` only makes sense when $a>0$`a`>0 and $b>0$`b`>0, $b\ne1$`b`≠1

Express $6^2=36$62=36 in logarithmic form

Think: Remember we want the index on its own and the logarithm on the other side

Do:

$6$6 is the base and $2$2 is the index, so:

$\log_636=2$`l``o``g`636=2

Rewrite in exponential form: $\log_432=2.5$`l``o``g`432=2.5

Think: Remember that in exponential form we want isolate the resulting number after having the base raised to the index

Do:

$32$32 is the result after raising the base $4$4 to the index $2.5$2.5, so:

$4^{2.5}=32$42.5=32

Evaluate $\log_216$`l``o``g`216.

Manipulate rational, exponential, and logarithmic algebraic expressions

Apply algebraic methods in solving problems