Write as a Single Logarithm log of 72-1/3(2* log of 4+ log of 32)

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

Step 1

Simplify each term.

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

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

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

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

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

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

Multiply $16$ by $32$ .

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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:

$0.95424250 \dots$