Exponential to Logarithmic Form

Exponential functions are the inverse of logarithmic functions. Put another way:

$b^y=x$ if and only if $y={\log }_{b}x$ for all $x>0,b>0,$ and $b\ne 1$

Since all logarithms are exponents, we can always express them using the same terminology. In this article, we'll practice rewriting exponential functions in logarithmic form. Let's get started!

Exponential to logarithmic form, explained

The easiest way to do this is to look at a few sample problems, so let's begin with an easy one. What is $2^3=8$ in logarithmic form?

Writing functions in logarithmic form means writing it in ${\log }_{b}a=x$ format where b is the base, a is the number after the = sign in the exponential equation, and x is the exponent in the exponential equation. In our sample problem above, $b=2,a=8,$ , and $x=3$ . Therefore, the logarithmic form of the equation would be:

${\log }_{2}8=3$

If you cannot remember which number goes where, a is the answer to the original equation and x stands for exponent. The b value is the base or the last number left. Let's tackle another one: $c^a=d$ .

This one is comprised exclusively of variables, but that doesn't change how we work with it. The c is our base or b value, a is our exponent x, and d is our answer on the other side of the = sign. Just be careful with the a since the one in the equation DOES NOT correspond to the a in ${\log }_{b}a=x$ . Our answer is:

${\log }_{c}d=a$

We run into a similar problem when the exponential function has an x value that isn't the exponent, giving us two different x values to think about. Consider the following example:

$8^4=x$

In this example, 4 is the exponent and therefore the x value for ${\log }_{b}a=x$ . The x in the equation corresponds to a in ${\log }_{b}a=x$ since it is the answer in the original equation. The 8 is our base or b value. Therefore, our logarithmic equation is:

${\log }_{8}x=4$

It's not that hard as long as we remember what all of the variables we're working with represent.

Practice questions on exponential to logarithmic form

a. Write in logarithmic form: $5^2=25$

Converting this expression to logarithmic form means putting it into ${\log }_{b}a=x$ format. In this example, 5 is the base or b value, 2 is the exponent or x value, and 25 is the answer or a value. Now, we simply plug in the values and get:

${\log }_{5}25=2$

b. Write in logarithmic form: $9^2=81$

Converting this expression to logarithmic form means putting it into ${\log }_{b}a=x$ format. In this example, 9 is the base or b value, 2 is the exponent or x value, and 81 is the answer or a value. Now, we simply plug in the values and get:

${\log }_{9}81=2$

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