Precalculus Examples

$4 \log_{4} (81)$

Step 1

Simplify $4 \log_{4} (81)$ by moving $4$ inside the logarithm.

$\log_{4} (81^{4})$

Step 2

Raise $81$ to the power of $4$ .

$\log_{4} (43046721)$

Step 3

Rewrite $\log_{4} (43046721)$ using the change of base formula.

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Step 3.1

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 3.2

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

$\frac{\log (43046721)}{\log (4)}$

Step 4

Rewrite $\log (43046721)$ as $\log (3^{16})$ .

$\frac{\log (3^{16})}{\log (4)}$

Step 5

Expand $\log (3^{16})$ by moving $16$ outside the logarithm.

$\frac{16 \log (3)}{\log (4)}$

Step 6

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{16 \log (3)}{\log (2^{2})}$

Step 7

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

$\frac{16 \log (3)}{2 \log (2)}$

Step 8

Cancel the common factor of $16$ and $2$ .

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Step 8.1

Factor $2$ out of $16 \log (3)$ .

$\frac{2 (8 \log (3))}{2 \log (2)}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $2 \log (2)$ .

$\frac{2 (8 \log (3))}{2 (\log (2))}$

Step 8.2.2

Cancel the common factor.

$\frac{2 (8 \log (3))}{2 \log (2)}$

Step 8.2.3

Rewrite the expression.

$\frac{8 \log (3)}{\log (2)}$

Step 9

Simplify $8 \log (3)$ by moving $8$ inside the logarithm.

$\frac{\log (3^{8})}{\log (2)}$

Step 10

Raise $3$ to the power of $8$ .

$\frac{\log (6561)}{\log (2)}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (6561)}{\log (2)}$

Decimal Form:

$12.67970000 \dots$