Precalculus Examples

$\log_{9} (49)$

Step 1

Rewrite $\log_{9} (49)$ using the change of base formula.

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Step 1.1

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 1.2

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

$\frac{\log (49)}{\log (9)}$

Step 2

Rewrite $\log (49)$ as $\log (7^{2})$ .

$\frac{\log (7^{2})}{\log (9)}$

Step 3

Expand $\log (7^{2})$ by moving $2$ outside the logarithm.

$\frac{2 \log (7)}{\log (9)}$

Step 4

Rewrite $\log (9)$ as $\log (3^{2})$ .

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

Step 5

Expand $\log (3^{2})$ by moving $2$ outside the logarithm.

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

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{\log (7)}{\log (3)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (7)}{\log (3)}$

Decimal Form:

$1.77124374 \dots$