Precalculus Examples

$13^{- 8 x} + 9 = 15$

Step 1

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

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Step 1.1

Subtract $9$ from both sides of the equation.

$13^{- 8 x} = 15 - 9$

Step 1.2

Subtract $9$ from $15$ .

$13^{- 8 x} = 6$

Step 2

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

$\ln (13^{- 8 x}) = \ln (6)$

Step 3

Expand $\ln (13^{- 8 x})$ by moving $- 8 x$ outside the logarithm.

$- 8 x \ln (13) = \ln (6)$

Step 4

Divide each term in $- 8 x \ln (13) = \ln (6)$ by $- 8 \ln (13)$ and simplify.

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Step 4.1

Divide each term in $- 8 x \ln (13) = \ln (6)$ by $- 8 \ln (13)$ .

$\frac{- 8 x \ln (13)}{- 8 \ln (13)} = \frac{\ln (6)}{- 8 \ln (13)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 8$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 8 x \ln (13)}{- 8 \ln (13)} = \frac{\ln (6)}{- 8 \ln (13)}$

Step 4.2.1.2

Rewrite the expression.

$\frac{x \ln (13)}{\ln (13)} = \frac{\ln (6)}{- 8 \ln (13)}$

Step 4.2.2

Cancel the common factor of $\ln (13)$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{x \ln (13)}{\ln (13)} = \frac{\ln (6)}{- 8 \ln (13)}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (6)}{- 8 \ln (13)}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (6)}{8 \ln (13)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (6)}{8 \ln (13)}$

Decimal Form:

$x = - 0.08731943 \dots$