Precalculus Examples

$\log_{5} (11 \cdot 10 \cdot 7^{5})$

Step 1

Rewrite $\log_{5} (11 \cdot 10 \cdot 7^{5})$ as $\log_{5} (11 \cdot 10) + \log_{5} (7^{5})$ .

$\log_{5} (11 \cdot 10) + \log_{5} (7^{5})$

Step 2

Rewrite $\log_{5} (11 \cdot 10)$ as $\log_{5} (11) + \log_{5} (10)$ .

$\log_{5} (11) + \log_{5} (10) + \log_{5} (7^{5})$

Step 3

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

$\log_{5} (11) + \log_{5} (10) + 5 \log_{5} (7)$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\log_{5} (10)$ as $\log_{5} (2 (5))$ .

$\log_{5} (11) + \log_{5} (2 (5)) + 5 \log_{5} (7)$

Step 4.2

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

$\log_{5} (11) + \log_{5} (2) + \log_{5} (5) + 5 \log_{5} (7)$

Step 4.3

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

$\log_{5} (11) + \log_{5} (2) + 1 + 5 \log_{5} (7)$