Precalculus Examples

$\frac{1}{2} \cdot \log (16) - \frac{1}{3} \cdot \log (8 + 1)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log (16)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (16^{\frac{1}{2}}) - \frac{1}{3} \cdot \log (8 + 1)$

Step 1.2

Rewrite $16$ as $4^{2}$ .

$\log ({(4^{2})}^{\frac{1}{2}}) - \frac{1}{3} \cdot \log (8 + 1)$

Step 1.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (4^{2 (\frac{1}{2})}) - \frac{1}{3} \cdot \log (8 + 1)$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\log (4^{2 (\frac{1}{2})}) - \frac{1}{3} \cdot \log (8 + 1)$

Step 1.4.2

Rewrite the expression.

$\log (4^{1}) - \frac{1}{3} \cdot \log (8 + 1)$

Step 1.5

Evaluate the exponent.

$\log (4) - \frac{1}{3} \cdot \log (8 + 1)$

Step 1.6

Add $8$ and $1$ .

$\log (4) - \frac{1}{3} \cdot \log (9)$

Step 1.7

Multiply $- \frac{1}{3} \log (9)$ .

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Step 1.7.1

Reorder $\log (9)$ and $\frac{1}{3}$ .

$\log (4) - (\frac{1}{3} \log (9))$

Step 1.7.2

Simplify $\frac{1}{3} \log (9)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log (4) - \log (9^{\frac{1}{3}})$

Step 2

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

$\log (\frac{4}{9^{\frac{1}{3}}})$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\log (\frac{4}{9^{\frac{1}{3}}})$

Decimal Form:

$0.28397915 \dots$