Precalculus Examples

$3^{x + 2} = 20$

Step 1

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

$\ln (3^{x + 2}) = \ln (20)$

Step 2

Expand $\ln (3^{x + 2})$ by moving $x + 2$ outside the logarithm.

$(x + 2) \ln (3) = \ln (20)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$x \ln (3) + 2 \ln (3) = \ln (20)$

Step 4

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

$x \ln (3) + 2 \ln (3) - \ln (20) = 0$

Step 5

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

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Step 5.1

Subtract $2 \ln (3)$ from both sides of the equation.

$x \ln (3) - \ln (20) = - 2 \ln (3)$

Step 5.2

Add $\ln (20)$ to both sides of the equation.

$x \ln (3) = - 2 \ln (3) + \ln (20)$

Step 6

Divide each term in $x \ln (3) = - 2 \ln (3) + \ln (20)$ by $\ln (3)$ and simplify.

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Step 6.1

Divide each term in $x \ln (3) = - 2 \ln (3) + \ln (20)$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{- 2 \ln (3)}{\ln (3)} + \frac{\ln (20)}{\ln (3)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\ln (3)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{- 2 \ln (3)}{\ln (3)} + \frac{\ln (20)}{\ln (3)}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (3)}{\ln (3)} + \frac{\ln (20)}{\ln (3)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $\ln (3)$ .

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Step 6.3.1.1

Cancel the common factor.

$x = \frac{- 2 \ln (3)}{\ln (3)} + \frac{\ln (20)}{\ln (3)}$

Step 6.3.1.2

Divide $- 2$ by $1$ .

$x = - 2 + \frac{\ln (20)}{\ln (3)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 2 + \frac{\ln (20)}{\ln (3)}$

Decimal Form:

$x = 0.72683302 \dots$