Precalculus Examples

$\log_{7} (\frac{\sqrt[3]{4 x^{5}}}{49 d})$

Step 1

Rewrite $\log_{7} (\frac{\sqrt[3]{4 x^{5}}}{49 d})$ as $\log_{7} (\sqrt[3]{4 x^{5}}) - \log_{7} (49 d)$ .

$\log_{7} (\sqrt[3]{4 x^{5}}) - \log_{7} (49 d)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x^{5}}$ as ${(4 x^{5})}^{\frac{1}{3}}$ .

$\log_{7} ({(4 x^{5})}^{\frac{1}{3}}) - \log_{7} (49 d)$

Step 3

Expand $\log_{7} ({(4 x^{5})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{7} (4 x^{5}) - \log_{7} (49 d)$

Step 4

Rewrite $\log_{7} (4 x^{5})$ as $\log_{7} (4) + \log_{7} (x^{5})$ .

$\frac{1}{3} (\log_{7} (4) + \log_{7} (x^{5})) - \log_{7} (49 d)$

Step 5

Expand $\log_{7} (x^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{3} (\log_{7} (4) + 5 \log_{7} (x)) - \log_{7} (49 d)$

Step 6

Rewrite $\log_{7} (49 d)$ as $\log_{7} (49) + \log_{7} (d)$ .

$\frac{1}{3} (\log_{7} (4) + 5 \log_{7} (x)) - (\log_{7} (49) + \log_{7} (d))$

Step 7

Logarithm base $7$ of $49$ is $2$ .

$\frac{1}{3} (\log_{7} (4) + 5 \log_{7} (x)) - (2 + \log_{7} (d))$

Step 8

Simplify each term.

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Step 8.1

Simplify each term.

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Step 8.1.1

Rewrite $\log_{7} (4)$ as $\log_{7} (2^{2})$ .

$\frac{1}{3} (\log_{7} (2^{2}) + 5 \log_{7} (x)) - (2 + \log_{7} (d))$

Step 8.1.2

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

$\frac{1}{3} (2 \log_{7} (2) + 5 \log_{7} (x)) - (2 + \log_{7} (d))$

Step 8.2

Apply the distributive property.

$\frac{1}{3} (2 \log_{7} (2)) + \frac{1}{3} (5 \log_{7} (x)) - (2 + \log_{7} (d))$

Step 8.3

Multiply $\frac{1}{3} (2 \log_{7} (2))$ .

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Step 8.3.1

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} \log_{7} (2) + \frac{1}{3} (5 \log_{7} (x)) - (2 + \log_{7} (d))$

Step 8.3.2

Combine $\frac{2}{3}$ and $\log_{7} (2)$ .

$\frac{2 \log_{7} (2)}{3} + \frac{1}{3} (5 \log_{7} (x)) - (2 + \log_{7} (d))$

Step 8.4

Multiply $\frac{1}{3} (5 \log_{7} (x))$ .

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Step 8.4.1

Combine $5$ and $\frac{1}{3}$ .

$\frac{2 \log_{7} (2)}{3} + \frac{5}{3} \log_{7} (x) - (2 + \log_{7} (d))$

Step 8.4.2

Combine $\frac{5}{3}$ and $\log_{7} (x)$ .

$\frac{2 \log_{7} (2)}{3} + \frac{5 \log_{7} (x)}{3} - (2 + \log_{7} (d))$

Step 8.5

Apply the distributive property.

$\frac{2 \log_{7} (2)}{3} + \frac{5 \log_{7} (x)}{3} - 1 \cdot 2 - \log_{7} (d)$

Step 8.6

Multiply $- 1$ by $2$ .

$\frac{2 \log_{7} (2)}{3} + \frac{5 \log_{7} (x)}{3} - 2 - \log_{7} (d)$