Simplify/Condense 2 log base 5 of 3-1/3 log base 5 of 64+ log base 5 of 16

$2 \log_{5} (3) - \frac{1}{3} \log_{5} (64) + \log_{5} (16)$

Step 1

Simplify each term.

Simplify $2 \log_{5} (3)$ by moving $2$ inside the logarithm.

$\log_{5} (3^{2}) - \frac{1}{3} \cdot \log_{5} (64) + \log_{5} (16)$

Raise $3$ to the power of $2$ .

$\log_{5} (9) - \frac{1}{3} \cdot \log_{5} (64) + \log_{5} (16)$

Multiply $- \frac{1}{3} \log_{5} (64)$ .

Reorder $\log_{5} (64)$ and $\frac{1}{3}$ .

$\log_{5} (9) - (\frac{1}{3} \log_{5} (64)) + \log_{5} (16)$

Simplify $\frac{1}{3} \log_{5} (64)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{5} (9) - \log_{5} (64^{\frac{1}{3}}) + \log_{5} (16)$

Rewrite $64$ as $4^{3}$ .

$\log_{5} (9) - \log_{5} ({(4^{3})}^{\frac{1}{3}}) + \log_{5} (16)$

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

$\log_{5} (9) - \log_{5} (4^{3 (\frac{1}{3})}) + \log_{5} (16)$

Cancel the common factor of $3$ .

Cancel the common factor.

$\log_{5} (9) - \log_{5} (4^{3 (\frac{1}{3})}) + \log_{5} (16)$

Rewrite the expression.

$\log_{5} (9) - \log_{5} (4^{1}) + \log_{5} (16)$

Evaluate the exponent.

$\log_{5} (9) - \log_{5} (4) + \log_{5} (16)$

Step 2

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

$\log_{5} (\frac{9}{4}) + \log_{5} (16)$

Step 3

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

$\log_{5} (\frac{9}{4} \cdot 16)$

Step 4

Cancel the common factor of $4$ .

Factor $4$ out of $16$ .

$\log_{5} (\frac{9}{4} \cdot (4 (4)))$

Cancel the common factor.

$\log_{5} (\frac{9}{4} \cdot (4 \cdot 4))$

Rewrite the expression.

$\log_{5} (9 \cdot 4)$

Step 5

Multiply $9$ by $4$ .

$\log_{5} (36)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\log_{5} (36)$

Decimal Form:

$2.22656550 \dots$

Simplify/Condense 2 log base 5 of 3-1/3 log base 5 of 64+ log base 5 of 16

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