Calculus Examples

$\log_{8} ({(\frac{2 \sqrt{x}}{5})}^{3})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\log_{8} ({(\frac{2 x^{\frac{1}{2}}}{5})}^{3})$

Step 2

Expand $\log_{8} ({(\frac{2 x^{\frac{1}{2}}}{5})}^{3})$ by moving $3$ outside the logarithm.

$3 \log_{8} (\frac{2 x^{\frac{1}{2}}}{5})$

Step 3

Rewrite $\log_{8} (\frac{2 x^{\frac{1}{2}}}{5})$ as $\log_{8} (2 x^{\frac{1}{2}}) - \log_{8} (5)$ .

$3 (\log_{8} (2 x^{\frac{1}{2}}) - \log_{8} (5))$

Step 4

Rewrite $\log_{8} (2 x^{\frac{1}{2}})$ as $\log_{8} (2) + \log_{8} (x^{\frac{1}{2}})$ .

$3 (\log_{8} (2) + \log_{8} (x^{\frac{1}{2}}) - \log_{8} (5))$

Step 5

Expand $\log_{8} (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$3 (\log_{8} (2) + \frac{1}{2} \log_{8} (x) - \log_{8} (5))$

Step 6

Logarithm base $8$ of $2$ is $\frac{1}{3}$ .

$3 (\frac{1}{3} + \frac{1}{2} \log_{8} (x) - \log_{8} (5))$

Step 7

Combine $\frac{1}{2}$ and $\log_{8} (x)$ .

$3 (\frac{1}{3} + \frac{\log_{8} (x)}{2} - \log_{8} (5))$

Step 8

Apply the distributive property.

$3 (\frac{1}{3}) + 3 \frac{\log_{8} (x)}{2} + 3 (- \log_{8} (5))$

Step 9

Simplify.

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Step 9.1

Cancel the common factor of $3$ .

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Step 9.1.1

Cancel the common factor.

$3 (\frac{1}{3}) + 3 \frac{\log_{8} (x)}{2} + 3 (- \log_{8} (5))$

Step 9.1.2

Rewrite the expression.

$1 + 3 \frac{\log_{8} (x)}{2} + 3 (- \log_{8} (5))$

Step 9.2

Combine $3$ and $\frac{\log_{8} (x)}{2}$ .

$1 + \frac{3 \log_{8} (x)}{2} + 3 (- \log_{8} (5))$

Step 9.3

Multiply $- 1$ by $3$ .

$1 + \frac{3 \log_{8} (x)}{2} - 3 \log_{8} (5)$