NZ Level 7 (NZC) Level 2 (NCEA)
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Logs and Exponential Forms

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:


Then we can rewrite it as the logarithm:


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).


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:

$\log_ba$logba only makes sense when $a>0$a>0 and $b>0$b>0$b\ne1$b1



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


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



question 2

Rewrite in exponential form: $\log_432=2.5$log432=2.5

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


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



question 3

Evaluate $\log_216$log216.




Manipulate rational, exponential, and logarithmic algebraic expressions


Apply algebraic methods in solving problems

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