Precalculus Examples

$5^{x} + 3 = 12$

Step 1

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

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Step 1.1

Subtract $3$ from both sides of the equation.

$5^{x} = 12 - 3$

Step 1.2

Subtract $3$ from $12$ .

$5^{x} = 9$

Step 2

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

$\ln (5^{x}) = \ln (9)$

Step 3

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

$x \ln (5) = \ln (9)$

Step 4

Divide each term in $x \ln (5) = \ln (9)$ by $\ln (5)$ and simplify.

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Step 4.1

Divide each term in $x \ln (5) = \ln (9)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (9)}{\ln (5)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (9)}{\ln (5)}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (9)}{\ln (5)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (9)}{\ln (5)}$

Decimal Form:

$x = 1.36521238 \dots$