Algebra Examples

$\frac{\log_{8} (2) + \log_{81} (3)}{\log_{\sqrt{3}} (27)}$

Step 1

Simplify the numerator.

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Step 1.1

Logarithm base $8$ of $2$ is $\frac{1}{3}$ .

$\frac{\frac{1}{3} + \log_{81} (3)}{\log_{\sqrt{3}} (27)}$

Step 1.2

Logarithm base $81$ of $3$ is $\frac{1}{4}$ .

$\frac{\frac{1}{3} + \frac{1}{4}}{\log_{\sqrt{3}} (27)}$

Step 1.3

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4}}{\log_{\sqrt{3}} (27)}$

Step 1.4

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{3}{3}}{\log_{\sqrt{3}} (27)}$

Step 1.5

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$\frac{\frac{4}{3 \cdot 4} + \frac{1}{4} \cdot \frac{3}{3}}{\log_{\sqrt{3}} (27)}$

Step 1.5.2

Multiply $3$ by $4$ .

$\frac{\frac{4}{12} + \frac{1}{4} \cdot \frac{3}{3}}{\log_{\sqrt{3}} (27)}$

Step 1.5.3

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\frac{\frac{4}{12} + \frac{3}{4 \cdot 3}}{\log_{\sqrt{3}} (27)}$

Step 1.5.4

Multiply $4$ by $3$ .

$\frac{\frac{4}{12} + \frac{3}{12}}{\log_{\sqrt{3}} (27)}$

Step 1.6

Combine the numerators over the common denominator.

$\frac{\frac{4 + 3}{12}}{\log_{\sqrt{3}} (27)}$

Step 1.7

Add $4$ and $3$ .

$\frac{\frac{7}{12}}{\log_{\sqrt{3}} (27)}$

Step 2

Logarithm base $\sqrt{3}$ of $27$ is $6$ .

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Step 2.1

Rewrite as an equation.

$\frac{\frac{7}{12}}{\log_{\sqrt{3}} (27) = x}$

Step 2.2

Rewrite $\log_{\sqrt{3}} (27) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$\frac{\frac{7}{12}}{{\sqrt{3}}^{x} = 27}$

Step 2.3

Create equivalent expressions in the equation that all have equal bases.

$\frac{\frac{7}{12}}{{\sqrt{3}}^{x} = {\sqrt{3}}^{6}}$

Step 2.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$\frac{\frac{7}{12}}{x = 6}$

Step 2.5

The variable $x$ is equal to $6$ .

$\frac{\frac{7}{12}}{6}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{7}{12} \cdot \frac{1}{6}$

Step 4

Multiply $\frac{7}{12} \cdot \frac{1}{6}$ .

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Step 4.1

Multiply $\frac{7}{12}$ by $\frac{1}{6}$ .

$\frac{7}{12 \cdot 6}$

Step 4.2

Multiply $12$ by $6$ .

$\frac{7}{72}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{72}$

Decimal Form:

$0.097\overline{2}$