Algebra Examples

$\log_{4} (5^{4})$

Step 1

Raise $5$ to the power of $4$ .

$\log_{4} (625)$

Step 2

Rewrite $\log_{4} (625)$ using the change of base formula.

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Step 2.1

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 2.2

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

$\frac{\log (625)}{\log (4)}$

Step 3

Rewrite $\log (625)$ as $\log (5^{4})$ .

$\frac{\log (5^{4})}{\log (4)}$

Step 4

Expand $\log (5^{4})$ by moving $4$ outside the logarithm.

$\frac{4 \log (5)}{\log (4)}$

Step 5

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

$\frac{4 \log (5)}{\log (2^{2})}$

Step 6

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

$\frac{4 \log (5)}{2 \log (2)}$

Step 7

Cancel the common factor of $4$ and $2$ .

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Step 7.1

Factor $2$ out of $4 \log (5)$ .

$\frac{2 (2 \log (5))}{2 \log (2)}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

$\frac{2 (2 \log (5))}{2 (\log (2))}$

Step 7.2.2

Cancel the common factor.

$\frac{2 (2 \log (5))}{2 \log (2)}$

Step 7.2.3

Rewrite the expression.

$\frac{2 \log (5)}{\log (2)}$

Step 8

Simplify $2 \log (5)$ by moving $2$ inside the logarithm.

$\frac{\log (5^{2})}{\log (2)}$

Step 9

Raise $5$ to the power of $2$ .

$\frac{\log (25)}{\log (2)}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (25)}{\log (2)}$

Decimal Form:

$4.64385618 \dots$