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# Exponential to Logarithmic Form

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

${b}^{y}=x$ if and only if $y={\mathrm{log}}_{b}x$ 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 ${\mathrm{log}}_{b}a=x$ 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:

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

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 ${\mathrm{log}}_{b}a=x$ . Our answer is:

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

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 ${\mathrm{log}}_{b}a=x$ . The x in the equation corresponds to a in ${\mathrm{log}}_{b}a=x$ since it is the answer in the original equation. The 8 is our base or b value. Therefore, our logarithmic equation is:

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

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 ${\mathrm{log}}_{b}a=x$ 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:

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

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

Converting this expression to logarithmic form means putting it into ${\mathrm{log}}_{b}a=x$ 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:

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

## Flashcards covering the Exponential to Logarithmic Form

Algebra II Flashcards

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