Logarithmic Functions

NZ Level 7 (NZC) Level 2 (NCEA)

Logs and Exponential Forms

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