Trigonometry Examples

$\log_{6} (36) = 5 x + 3$

Step 1

Rewrite the equation as $5 x + 3 = \log_{6} (36)$ .

$5 x + 3 = \log_{6} (36)$

Step 2

Logarithm base $6$ of $36$ is $2$ .

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Step 2.1

Rewrite as an equation.

$5 x + 3 = \log_{6} (36) = x$

Step 2.2

Rewrite $\log_{6} (36) = x$ 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 $b^{y} = x$ .

$5 x + 3 = 6^{x} = 36$

Step 2.3

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

$5 x + 3 = 6^{x} = 6^{2}$

Step 2.4

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

$5 x + 3 = x = 2$

Step 2.5

The variable $x$ is equal to $2$ .

$5 x + 3 = 2$

Step 3

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

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Step 3.1

Subtract $3$ from both sides of the equation.

$5 x = 2 - 3$

Step 3.2

Subtract $3$ from $2$ .

$5 x = - 1$

Step 4

Divide each term in $5 x = - 1$ by $5$ and simplify.

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Step 4.1

Divide each term in $5 x = - 1$ by $5$ .

$\frac{5 x}{5} = \frac{- 1}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 1}{5}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{5}$

Decimal Form:

$x = - 0.2$