Trigonometry Examples

$\log_{8} (16) = x + 1$

Step 1

Rewrite the equation as $x + 1 = \log_{8} (16)$ .

$x + 1 = \log_{8} (16)$

Step 2

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

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Step 2.1

Rewrite as an equation.

$x + 1 = \log_{8} (16) = x$

Step 2.2

Rewrite $\log_{8} (16) = x$ 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 $b^{y} = x$ .

$x + 1 = 8^{x} = 16$

Step 2.3

Create expressions in the equation that all have equal bases.

$x + 1 = {(2^{3})}^{x} = 2^{4}$

Step 2.4

Rewrite ${(2^{3})}^{x}$ as $2^{3 x}$ .

$x + 1 = 2^{3 x} = 2^{4}$

Step 2.5

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

$x + 1 = 3 x = 4$

Step 2.6

Solve for $x$ .

$x + 1 = x = \frac{4}{3}$

Step 2.7

The variable $x$ is equal to $\frac{4}{3}$ .

$x + 1 = \frac{4}{3}$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Subtract $1$ from both sides of the equation.

$x = \frac{4}{3} - 1$

Step 3.2

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

$x = \frac{4}{3} - 1 \cdot \frac{3}{3}$

Step 3.3

Combine $- 1$ and $\frac{3}{3}$ .

$x = \frac{4}{3} + \frac{- 1 \cdot 3}{3}$

Step 3.4

Combine the numerators over the common denominator.

$x = \frac{4 - 1 \cdot 3}{3}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $3$ .

$x = \frac{4 - 3}{3}$

Step 3.5.2

Subtract $3$ from $4$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{3}$