Algebra Examples

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

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.

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Step 2.1

Divide $625$ by $125$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 2.6

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:

$3.90689059 \dots$