Trigonometry Examples

$\log_{11} (x - 7) = - 5$

Step 1

Rewrite $\log_{11} (x - 7) = - 5$ 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 $b^{y} = x$ .

$11^{- 5} = x - 7$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x - 7 = 11^{- 5}$ .

$x - 7 = 11^{- 5}$

Step 2.2

Simplify $11^{- 5}$ .

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Step 2.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x - 7 = \frac{1}{11^{5}}$

Step 2.2.2

Raise $11$ to the power of $5$ .

$x - 7 = \frac{1}{161051}$

Step 2.3

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

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Step 2.3.1

Add $7$ to both sides of the equation.

$x = \frac{1}{161051} + 7$

Step 2.3.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{161051}{161051}$ .

$x = \frac{1}{161051} + 7 \cdot \frac{161051}{161051}$

Step 2.3.3

Combine $7$ and $\frac{161051}{161051}$ .

$x = \frac{1}{161051} + \frac{7 \cdot 161051}{161051}$

Step 2.3.4

Combine the numerators over the common denominator.

$x = \frac{1 + 7 \cdot 161051}{161051}$

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $7$ by $161051$ .

$x = \frac{1 + 1127357}{161051}$

Step 2.3.5.2

Add $1$ and $1127357$ .

$x = \frac{1127358}{161051}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1127358}{161051}$

Decimal Form:

$x = 7.00000620 \dots$

Mixed Number Form:

$x = 7 \frac{1}{161051}$