Precalculus Examples

$\log (8) + 3 \log (5) - \frac{1}{6} \cdot \log (m) + \log (n)$

Step 1

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

$3 \log (5) - \frac{1}{6} \cdot \log (m) + \log (8 n)$

Step 2

Simplify each term.

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Step 2.1

Simplify $3 \log (5)$ by moving $3$ inside the logarithm.

$\log (5^{3}) - \frac{1}{6} \cdot \log (m) + \log (8 n)$

Step 2.2

Raise $5$ to the power of $3$ .

$\log (125) - \frac{1}{6} \cdot \log (m) + \log (8 n)$

Step 2.3

Multiply $- \frac{1}{6} \log (m)$ .

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Step 2.3.1

Reorder $\log (m)$ and $\frac{1}{6}$ .

$\log (125) - (\frac{1}{6} \log (m)) + \log (8 n)$

Step 2.3.2

Simplify $\frac{1}{6} \log (m)$ by moving $\frac{1}{6}$ inside the logarithm.

$\log (125) - \log (m^{\frac{1}{6}}) + \log (8 n)$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{125}{m^{\frac{1}{6}}}) + \log (8 n)$

Step 4

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

$\log (\frac{125}{m^{\frac{1}{6}}} (8 n))$

Step 5

Rewrite using the commutative property of multiplication.

$\log (8 \frac{125}{m^{\frac{1}{6}}} n)$

Step 6

Multiply $8 \frac{125}{m^{\frac{1}{6}}}$ .

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Step 6.1

Combine $8$ and $\frac{125}{m^{\frac{1}{6}}}$ .

$\log (\frac{8 \cdot 125}{m^{\frac{1}{6}}} n)$

Step 6.2

Multiply $8$ by $125$ .

$\log (\frac{1000}{m^{\frac{1}{6}}} n)$

Step 7

Combine $\frac{1000}{m^{\frac{1}{6}}}$ and $n$ .

$\log (\frac{1000 n}{m^{\frac{1}{6}}})$