Precalculus Examples

$e^{- x} = 125^{2 x + 3}$

Step 1

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

$\ln (e^{- x}) = \ln (125^{2 x + 3})$

Step 2

Expand the left side.

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Step 2.1

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

$- x \ln (e) = \ln (125^{2 x + 3})$

Step 2.2

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

$- x \cdot 1 = \ln (125^{2 x + 3})$

Step 2.3

Multiply $- 1$ by $1$ .

$- x = \ln (125^{2 x + 3})$

Step 3

Expand $\ln (125^{2 x + 3})$ by moving $2 x + 3$ outside the logarithm.

$- x = (2 x + 3) \ln (125)$

Step 4

Simplify the right side.

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Step 4.1

Apply the distributive property.

$- x = 2 x \ln (125) + 3 \ln (125)$

Step 5

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

$- 2 x \ln (125) - 3 \ln (125) = x$

Step 6

Subtract $x$ from both sides of the equation.

$- 2 x \ln (125) - 3 \ln (125) - x = 0$

Step 7

Add $3 \ln (125)$ to both sides of the equation.

$- 2 x \ln (125) - x = 3 \ln (125)$

Step 8

Factor $x$ out of $- 2 x \ln (125) - x$ .

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Step 8.1

Factor $x$ out of $- 2 x \ln (125)$ .

$x (- 2 \ln (125)) - x = 3 \ln (125)$

Step 8.2

Factor $x$ out of $- x$ .

$x (- 2 \ln (125)) + x \cdot - 1 = 3 \ln (125)$

Step 8.3

Factor $x$ out of $x (- 2 \ln (125)) + x \cdot - 1$ .

$x (- 2 \ln (125) - 1) = 3 \ln (125)$

Step 9

Divide each term in $x (- 2 \ln (125) - 1) = 3 \ln (125)$ by $- 2 \ln (125) - 1$ and simplify.

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Step 9.1

Divide each term in $x (- 2 \ln (125) - 1) = 3 \ln (125)$ by $- 2 \ln (125) - 1$ .

$\frac{x (- 2 \ln (125) - 1)}{- 2 \ln (125) - 1} = \frac{3 \ln (125)}{- 2 \ln (125) - 1}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 2 \ln (125) - 1$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{x (- 2 \ln (125) - 1)}{- 2 \ln (125) - 1} = \frac{3 \ln (125)}{- 2 \ln (125) - 1}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 \ln (125)}{- 2 \ln (125) - 1}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Factor $- 1$ out of $- 2 \ln (125)$ .

$x = \frac{3 \ln (125)}{- (2 \ln (125)) - 1}$

Step 9.3.2

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{3 \ln (125)}{- (2 \ln (125)) - 1 (1)}$

Step 9.3.3

Factor $- 1$ out of $- (2 \ln (125)) - 1 (1)$ .

$x = \frac{3 \ln (125)}{- (2 \ln (125) + 1)}$

Step 9.3.4

Rewrite negatives.

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Step 9.3.4.1

Rewrite $- (2 \ln (125) + 1)$ as $- 1 (2 \ln (125) + 1)$ .

$x = \frac{3 \ln (125)}{- 1 (2 \ln (125) + 1)}$

Step 9.3.4.2

Move the negative in front of the fraction.

$x = - \frac{3 \ln (125)}{2 \ln (125) + 1}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 \ln (125)}{2 \ln (125) + 1}$

Decimal Form:

$x = - 1.35924252 \dots$