Precalculus Examples

$5^{x + 8} = 4$

Step 1

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

$\ln (5^{x + 8}) = \ln (4)$

Step 2

Expand $\ln (5^{x + 8})$ by moving $x + 8$ outside the logarithm.

$(x + 8) \ln (5) = \ln (4)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$x \ln (5) + 8 \ln (5) = \ln (4)$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (5) + 8 \ln (5) - \ln (4) = 0$

Step 5

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

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Step 5.1

Subtract $8 \ln (5)$ from both sides of the equation.

$x \ln (5) - \ln (4) = - 8 \ln (5)$

Step 5.2

Add $\ln (4)$ to both sides of the equation.

$x \ln (5) = - 8 \ln (5) + \ln (4)$

Step 6

Divide each term in $x \ln (5) = - 8 \ln (5) + \ln (4)$ by $\ln (5)$ and simplify.

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Step 6.1

Divide each term in $x \ln (5) = - 8 \ln (5) + \ln (4)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{- 8 \ln (5)}{\ln (5)} + \frac{\ln (4)}{\ln (5)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\ln (5)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{- 8 \ln (5)}{\ln (5)} + \frac{\ln (4)}{\ln (5)}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8 \ln (5)}{\ln (5)} + \frac{\ln (4)}{\ln (5)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $\ln (5)$ .

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Step 6.3.1.1

Cancel the common factor.

$x = \frac{- 8 \ln (5)}{\ln (5)} + \frac{\ln (4)}{\ln (5)}$

Step 6.3.1.2

Divide $- 8$ by $1$ .

$x = - 8 + \frac{\ln (4)}{\ln (5)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 8 + \frac{\ln (4)}{\ln (5)}$

Decimal Form:

$x = - 7.13864688 \dots$