Algebra Examples

$\log (25) + \frac{\log (8) \cdot \log (16)}{\log (64)}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\log (25) + \frac{\log (2^{3}) \cdot \log (16)}{\log (64)}$

Step 1.2

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

$\log (25) + \frac{3 \log (2) \cdot \log (16)}{\log (64)}$

Step 1.3

Rewrite $\log (16)$ as $\log (2^{4})$ .

$\log (25) + \frac{3 \log (2) \cdot \log (2^{4})}{\log (64)}$

Step 1.4

Expand $\log (2^{4})$ by moving $4$ outside the logarithm.

$\log (25) + \frac{3 \log (2) \cdot (4 \log (2))}{\log (64)}$

Step 1.5

Rewrite $\log (64)$ as $\log (2^{6})$ .

$\log (25) + \frac{3 \log (2) \cdot (4 \log (2))}{\log (2^{6})}$

Step 1.6

Expand $\log (2^{6})$ by moving $6$ outside the logarithm.

$\log (25) + \frac{3 \log (2) \cdot (4 \log (2))}{6 \log (2)}$

Step 1.7

Cancel the common factor of $3$ and $6$ .

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Step 1.7.1

Factor $3$ out of $3 \log (2) \cdot (4 \log (2))$ .

$\log (25) + \frac{3 (\log (2) \cdot (4 \log (2)))}{6 \log (2)}$

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

Factor $3$ out of $6 \log (2)$ .

$\log (25) + \frac{3 (\log (2) \cdot (4 \log (2)))}{3 (2 \log (2))}$

Step 1.7.2.2

Cancel the common factor.

$\log (25) + \frac{3 (\log (2) \cdot (4 \log (2)))}{3 (2 \log (2))}$

Step 1.7.2.3

Rewrite the expression.

$\log (25) + \frac{\log (2) \cdot (4 \log (2))}{2 \log (2)}$

Step 1.8

Cancel the common factor of $\log (2)$ .

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Step 1.8.1

Cancel the common factor.

$\log (25) + \frac{\log (2) \cdot (4 \log (2))}{2 \log (2)}$

Step 1.8.2

Rewrite the expression.

$\log (25) + \frac{4 \log (2)}{2}$

Step 1.9

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

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Step 1.9.1

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

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

Step 1.9.2

Cancel the common factors.

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Step 1.9.2.1

Factor $2$ out of $2$ .

$\log (25) + \frac{2 (2 \log (2))}{2 (1)}$

Step 1.9.2.2

Cancel the common factor.

$\log (25) + \frac{2 (2 \log (2))}{2 \cdot 1}$

Step 1.9.2.3

Rewrite the expression.

$\log (25) + \frac{2 \log (2)}{1}$

Step 1.9.2.4

Divide $2 \log (2)$ by $1$ .

$\log (25) + 2 \log (2)$

Step 1.10

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

$\log (25) + \log (2^{2})$

Step 1.11

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

$\log (25) + \log (4)$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (25 \cdot 4)$

Step 3

Multiply $25$ by $4$ .

$\log (100)$

Step 4

Logarithm base $10$ of $100$ is $2$ .

$2$