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:
If we have an exponential of the form:
$b^x=a$bx=a
Then we can rewrite it as the logarithm:
$\log_ba=x$logba=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).
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, $b\ne1$b≠1
Examples
QUESTION 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:
$\log_636=2$log636=2
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
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
question 3
Evaluate $\log_216$log216.