Precalculus Examples

$\log_{5} (7 \sqrt[3]{10 \cdot 3})$

Step 1

Rewrite $\log_{5} (7 \sqrt[3]{10 \cdot 3})$ as $\log_{5} (7) + \log_{5} (\sqrt[3]{10 \cdot 3})$ .

$\log_{5} (7) + \log_{5} (\sqrt[3]{10 \cdot 3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{10 \cdot 3}$ as ${(10 \cdot 3)}^{\frac{1}{3}}$ .

$\log_{5} (7) + \log_{5} ({(10 \cdot 3)}^{\frac{1}{3}})$

Step 3

Expand $\log_{5} ({(10 \cdot 3)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log_{5} (7) + \frac{1}{3} \log_{5} (10 \cdot 3)$

Step 4

Multiply $10$ by $3$ .

$\log_{5} (7) + \frac{1}{3} \log_{5} (30)$

Step 5

Rewrite $\log_{5} (30)$ as $\log_{5} (2 \cdot 3 \cdot 5)$ .

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

Step 6

Rewrite $\log_{5} (2 \cdot 3 \cdot 5)$ as $\log_{5} (2 \cdot 3) + \log_{5} (5)$ .

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

Step 7

Multiply $2$ by $3$ .

$\log_{5} (7) + \frac{1}{3} (\log_{5} (6) + \log_{5} (5))$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\log_{5} (6)$ as $\log_{5} (2 (3))$ .

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

Step 8.2

Rewrite $\log_{5} (2 (3))$ as $\log_{5} (2) + \log_{5} (3)$ .

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

Step 8.3

Logarithm base $5$ of $5$ is $1$ .

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

Step 9

Apply the distributive property.

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

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{3}$ and $\log_{5} (2)$ .

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

Step 10.2

Combine $\frac{1}{3}$ and $\log_{5} (3)$ .

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

Step 10.3

Multiply $\frac{1}{3}$ by $1$ .

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