Precalculus Examples

$e^{y} = 9$ , $7^{- 6 x} = 11$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (7^{- 6 x}) = \ln (11)$

Step 2

Expand $\ln (7^{- 6 x})$ by moving $- 6 x$ outside the logarithm.

$- 6 x \ln (7) = \ln (11)$

Step 3

Divide each term in $- 6 x \ln (7) = \ln (11)$ by $- 6 \ln (7)$ and simplify.

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Step 3.1

Divide each term in $- 6 x \ln (7) = \ln (11)$ by $- 6 \ln (7)$ .

$\frac{- 6 x \ln (7)}{- 6 \ln (7)} = \frac{\ln (11)}{- 6 \ln (7)}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 6$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 6 x \ln (7)}{- 6 \ln (7)} = \frac{\ln (11)}{- 6 \ln (7)}$

Step 3.2.1.2

Rewrite the expression.

$\frac{x \ln (7)}{\ln (7)} = \frac{\ln (11)}{- 6 \ln (7)}$

Step 3.2.2

Cancel the common factor of $\ln (7)$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{x \ln (7)}{\ln (7)} = \frac{\ln (11)}{- 6 \ln (7)}$

Step 3.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (11)}{- 6 \ln (7)}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (11)}{6 \ln (7)}$

Step 4

Replace all occurrences of $x$ with $- \frac{\ln (11)}{6 \ln (7)}$ in each equation.

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Step 4.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{y}) = \ln (9)$

$x = - \frac{\ln (11)}{6 \ln (7)}$

Step 4.2

Expand the left side.

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Step 4.2.1

Expand $\ln (e^{y})$ by moving $y$ outside the logarithm.

$y \ln (e) = \ln (9)$

$x = - \frac{\ln (11)}{6 \ln (7)}$

Step 4.2.2

The natural logarithm of $e$ is $1$ .

$y \cdot 1 = \ln (9)$

$x = - \frac{\ln (11)}{6 \ln (7)}$

Step 4.2.3

Multiply $y$ by $1$ .

$y = \ln (9)$

$x = - \frac{\ln (11)}{6 \ln (7)}$

$y = \ln (9)$

$x = - \frac{\ln (11)}{6 \ln (7)}$

$y = \ln (9)$

$x = - \frac{\ln (11)}{6 \ln (7)}$

Step 5

Simplify the right side.

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Step 5.1

Simplify $- \frac{\ln (11)}{6 \ln (7)}$ .

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Step 5.1.1

Simplify $6 \ln (7)$ by moving $6$ inside the logarithm.

$x = - \frac{\ln (11)}{\ln (7^{6})}$

$y = \ln (9)$

Step 5.1.2

Raise $7$ to the power of $6$ .

$x = - \frac{\ln (11)}{\ln (117649)}$

$y = \ln (9)$

$x = - \frac{\ln (11)}{\ln (117649)}$

$y = \ln (9)$

$x = - \frac{\ln (11)}{\ln (117649)}$

$y = \ln (9)$