Algebra Examples

$\log (72) - \frac{1}{3} (2 \cdot \log (4) + \log (32))$

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify $2 \log (4)$ by moving $2$ inside the logarithm.

$\log (72) - \frac{1}{3} \cdot (\log (4^{2}) + \log (32))$

Step 1.1.2

Raise $4$ to the power of $2$ .

$\log (72) - \frac{1}{3} \cdot (\log (16) + \log (32))$

Step 1.2

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

$\log (72) - \frac{1}{3} \cdot \log (16 \cdot 32)$

Step 1.3

Multiply $16$ by $32$ .

$\log (72) - \frac{1}{3} \cdot \log (512)$

Step 1.4

Multiply $- \frac{1}{3} \log (512)$ .

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Step 1.4.1

Reorder $\log (512)$ and $\frac{1}{3}$ .

$\log (72) - (\frac{1}{3} \log (512))$

Step 1.4.2

Simplify $\frac{1}{3} \log (512)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log (72) - \log (512^{\frac{1}{3}})$

Step 1.5

Rewrite $512$ as $8^{3}$ .

$\log (72) - \log ({(8^{3})}^{\frac{1}{3}})$

Step 1.6

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

$\log (72) - \log (8^{3 (\frac{1}{3})})$

Step 1.7

Cancel the common factor of $3$ .

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Step 1.7.1

Cancel the common factor.

$\log (72) - \log (8^{3 (\frac{1}{3})})$

Step 1.7.2

Rewrite the expression.

$\log (72) - \log (8^{1})$

Step 1.8

Evaluate the exponent.

$\log (72) - \log (8)$

Step 2

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

$\log (\frac{72}{8})$

Step 3

Divide $72$ by $8$ .

$\log (9)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\log (9)$

Decimal Form:

$0.95424250 \dots$