Trigonometry Examples

$6 \log_{6} (x + 1) = 5$

Step 1

Divide each term in $6 \log_{6} (x + 1) = 5$ by $6$ and simplify.

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Step 1.1

Divide each term in $6 \log_{6} (x + 1) = 5$ by $6$ .

$\frac{6 \log_{6} (x + 1)}{6} = \frac{5}{6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $6$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{6 \log_{6} (x + 1)}{6} = \frac{5}{6}$

Step 1.2.1.2

Divide $\log_{6} (x + 1)$ by $1$ .

$\log_{6} (x + 1) = \frac{5}{6}$

Step 2

Rewrite $\log_{6} (x + 1) = \frac{5}{6}$ 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 $b^{y} = x$ .

$6^{\frac{5}{6}} = x + 1$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x + 1 = 6^{\frac{5}{6}}$ .

$x + 1 = 6^{\frac{5}{6}}$

Step 3.2

Subtract $1$ from both sides of the equation.

$x = 6^{\frac{5}{6}} - 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 6^{\frac{5}{6}} - 1$

Decimal Form:

$x = 3.45101825 \dots$