Simplify/Condense log base 2 of 625- log base 2 of 125+1/3 log base 2 of 27

$\log_{2} (625) - \log_{2} (125) + \frac{1}{3} \log_{2} (27)$

Step 1

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

$\log_{2} (\frac{625}{125}) + \frac{1}{3} \cdot \log_{2} (27)$

Step 2

Simplify each term.

Divide $625$ by $125$ .

$\log_{2} (5) + \frac{1}{3} \cdot \log_{2} (27)$

Simplify $\frac{1}{3} \log_{2} (27)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{2} (5) + \log_{2} (27^{\frac{1}{3}})$

Rewrite $27$ as $3^{3}$ .

$\log_{2} (5) + \log_{2} ({(3^{3})}^{\frac{1}{3}})$

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

$\log_{2} (5) + \log_{2} (3^{3 (\frac{1}{3})})$

Cancel the common factor of $3$ .

Cancel the common factor.

$\log_{2} (5) + \log_{2} (3^{3 (\frac{1}{3})})$

Rewrite the expression.

$\log_{2} (5) + \log_{2} (3^{1})$

Evaluate the exponent.

$\log_{2} (5) + \log_{2} (3)$

Step 3

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

$\log_{2} (5 \cdot 3)$

Step 4

Multiply $5$ by $3$ .

$\log_{2} (15)$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\log_{2} (15)$

Decimal Form:

$3.90689059 \dots$