Precalculus Examples

$\frac{\log_{3} (7)}{\log_{3} (9)}$

Step 1

Logarithm base $3$ of $9$ is $2$ .

$\frac{\log_{3} (7)}{2}$

Step 2

Rewrite $\frac{\log_{3} (7)}{2}$ as $\frac{1}{2} \log_{3} (7)$ .

$\frac{1}{2} \log_{3} (7)$

Step 3

Simplify $\frac{1}{2} \log_{3} (7)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{3} (7^{\frac{1}{2}})$

Step 4

Rewrite $\log_{3} (7^{\frac{1}{2}})$ using the change of base formula.

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Step 4.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 4.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (7^{\frac{1}{2}})}{\log (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (7^{\frac{1}{2}})}{\log (3)}$

Decimal Form:

$0.88562187 \dots$