Write as a Single Logarithm 1/3( log base 5 of 8+ log base 5 of 27)- log base 5 of 3

$\frac{1}{3} (\log_{5} (8) + \log_{5} (27)) - \log_{5} (3)$

Step 1

Simplify each term.

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

$\frac{1}{3} \cdot \log_{5} (8 \cdot 27) - \log_{5} (3)$

Multiply $8$ by $27$ .

$\frac{1}{3} \cdot \log_{5} (216) - \log_{5} (3)$

Simplify $\frac{1}{3} \log_{5} (216)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{5} (216^{\frac{1}{3}}) - \log_{5} (3)$

Rewrite $216$ as $6^{3}$ .

$\log_{5} ({(6^{3})}^{\frac{1}{3}}) - \log_{5} (3)$

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

$\log_{5} (6^{3 (\frac{1}{3})}) - \log_{5} (3)$

Cancel the common factor of $3$ .

Cancel the common factor.

$\log_{5} (6^{3 (\frac{1}{3})}) - \log_{5} (3)$

Rewrite the expression.

$\log_{5} (6^{1}) - \log_{5} (3)$

Evaluate the exponent.

$\log_{5} (6) - \log_{5} (3)$

Step 2

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

$\log_{5} (\frac{6}{3})$

Step 3

Divide $6$ by $3$ .

$\log_{5} (2)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\log_{5} (2)$

Decimal Form:

$0.43067655 \dots$