Algebra Examples

$\log_{8} (16) + 2 \log_{8} (x) = 2$

Step 1

Logarithm base $8$ of $16$ is $\frac{4}{3}$ .

$\frac{4}{3} + 2 \log_{8} (x) = 2$

Step 2

Move all terms not containing $\log_{8} (x)$ to the right side of the equation.

Tap for more steps...

Step 2.1

Subtract $\frac{4}{3}$ from both sides of the equation.

$2 \log_{8} (x) = 2 - \frac{4}{3}$

Step 2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 \log_{8} (x) = 2 \cdot \frac{3}{3} - \frac{4}{3}$

Step 2.3

Combine $2$ and $\frac{3}{3}$ .

$2 \log_{8} (x) = \frac{2 \cdot 3}{3} - \frac{4}{3}$

Step 2.4

Combine the numerators over the common denominator.

$2 \log_{8} (x) = \frac{2 \cdot 3 - 4}{3}$

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Multiply $2$ by $3$ .

$2 \log_{8} (x) = \frac{6 - 4}{3}$

Step 2.5.2

Subtract $4$ from $6$ .

$2 \log_{8} (x) = \frac{2}{3}$

Step 3

Divide each term in $2 \log_{8} (x) = \frac{2}{3}$ by $2$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $2 \log_{8} (x) = \frac{2}{3}$ by $2$ .

$\frac{2 \log_{8} (x)}{2} = \frac{\frac{2}{3}}{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{2 \log_{8} (x)}{2} = \frac{\frac{2}{3}}{2}$

Step 3.2.1.2

Divide $\log_{8} (x)$ by $1$ .

$\log_{8} (x) = \frac{\frac{2}{3}}{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$\log_{8} (x) = \frac{2}{3} \cdot \frac{1}{2}$

Step 3.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1

Cancel the common factor.

$\log_{8} (x) = \frac{2}{3} \cdot \frac{1}{2}$

Step 3.3.2.2

Rewrite the expression.

$\log_{8} (x) = \frac{1}{3}$

Step 4

Rewrite $\log_{8} (x) = \frac{1}{3}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$8^{\frac{1}{3}} = x$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $x = 8^{\frac{1}{3}}$ .

$x = 8^{\frac{1}{3}}$

Step 5.2

Simplify $8^{\frac{1}{3}}$ .

Tap for more steps...

Step 5.2.1

Simplify the expression.

Tap for more steps...

Step 5.2.1.1

Rewrite $8$ as $2^{3}$ .

$x = {(2^{3})}^{\frac{1}{3}}$

Step 5.2.1.2

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

$x = 2^{3 (\frac{1}{3})}$

Step 5.2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.2.1

Cancel the common factor.

$x = 2^{3 (\frac{1}{3})}$

Step 5.2.2.2

Rewrite the expression.

$x = 2^{1}$

Step 5.2.3

Evaluate the exponent.

$x = 2$