Precalculus Examples

$5 e^{0.5 x - 2} = {1.9}^{x}$

Step 1

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

$\ln (5 e^{0.5 x - 2}) = \ln ({1.9}^{x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (5 e^{0.5 x - 2})$ as $\ln (5) + \ln (e^{0.5 x - 2})$ .

$\ln (5) + \ln (e^{0.5 x - 2}) = \ln ({1.9}^{x})$

Step 2.2

Expand $\ln (e^{0.5 x - 2})$ by moving $0.5 x - 2$ outside the logarithm.

$\ln (5) + (0.5 x - 2) \ln (e) = \ln ({1.9}^{x})$

Step 2.3

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

$\ln (5) + (0.5 x - 2) \cdot 1 = \ln ({1.9}^{x})$

Step 2.4

Multiply $0.5 x - 2$ by $1$ .

$\ln (5) + 0.5 x - 2 = \ln ({1.9}^{x})$

Step 3

Expand $\ln ({1.9}^{x})$ by moving $x$ outside the logarithm.

$\ln (5) + 0.5 x - 2 = x \ln (1.9)$

Step 4

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

$\ln (5) - x \ln (1.9) = - 0.5 x + 2$

Step 5

Add $0.5 x$ to both sides of the equation.

$\ln (5) - x \ln (1.9) + 0.5 x = 2$

Step 6

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

$- x \ln (1.9) + 0.5 x = 2 - \ln (5)$

Step 7

Factor $0.5 x$ out of $- x \ln (1.9) + 0.5 x$ .

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Step 7.1

Factor $0.5 x$ out of $- x \ln (1.9)$ .

$0.5 x (- 2 \ln (1.9)) + 0.5 x = 2 - \ln (5)$

Step 7.2

Factor $0.5 x$ out of $0.5 x$ .

$0.5 x (- 2 \ln (1.9)) + 0.5 x (1) = 2 - \ln (5)$

Step 7.3

Factor $0.5 x$ out of $0.5 x (- 2 \ln (1.9)) + 0.5 x (1)$ .

$0.5 x (- 2 \ln (1.9) + 1) = 2 - \ln (5)$

Step 8

Divide each term in $0.5 x (- 2 \ln (1.9) + 1) = 2 - \ln (5)$ by $0.5 (- 2 \ln (1.9) + 1)$ and simplify.

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Step 8.1

Divide each term in $0.5 x (- 2 \ln (1.9) + 1) = 2 - \ln (5)$ by $0.5 (- 2 \ln (1.9) + 1)$ .

$\frac{0.5 x (- 2 \ln (1.9) + 1)}{0.5 (- 2 \ln (1.9) + 1)} = \frac{2}{0.5 (- 2 \ln (1.9) + 1)} + \frac{- \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $0.5$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{0.5 x (- 2 \ln (1.9) + 1)}{0.5 (- 2 \ln (1.9) + 1)} = \frac{2}{0.5 (- 2 \ln (1.9) + 1)} + \frac{- \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.2.1.2

Rewrite the expression.

$\frac{x (- 2 \ln (1.9) + 1)}{- 2 \ln (1.9) + 1} = \frac{2}{0.5 (- 2 \ln (1.9) + 1)} + \frac{- \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.2.2

Cancel the common factor of $- 2 \ln (1.9) + 1$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{x (- 2 \ln (1.9) + 1)}{- 2 \ln (1.9) + 1} = \frac{2}{0.5 (- 2 \ln (1.9) + 1)} + \frac{- \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.2.2.2

Divide $x$ by $1$ .

$x = \frac{2}{0.5 (- 2 \ln (1.9) + 1)} + \frac{- \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{0.5 (- 2 \ln (1.9) + 1)} + \frac{- \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.3.1.2

Separate fractions.

$x = \frac{2}{0.5} \cdot \frac{1}{- 2 \ln (1.9) + 1} + \frac{- \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.3.1.3

Divide $2$ by $0.5$ .

$x = 4 \frac{1}{- 2 \ln (1.9) + 1} + \frac{- \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.3.1.4

Combine $4$ and $\frac{1}{- 2 \ln (1.9) + 1}$ .

$x = \frac{4}{- 2 \ln (1.9) + 1} + \frac{- \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.3.1.5

Move the negative in front of the fraction.

$x = \frac{4}{- 2 \ln (1.9) + 1} - \frac{\ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.3.1.6

Multiply by $1$ .

$x = \frac{4}{- 2 \ln (1.9) + 1} - \frac{1 \ln (5)}{0.5 (- 2 \ln (1.9) + 1)}$

Step 8.3.1.7

Separate fractions.

$x = \frac{4}{- 2 \ln (1.9) + 1} - (\frac{1}{0.5} \cdot \frac{\ln (5)}{- 2 \ln (1.9) + 1})$

Step 8.3.1.8

Divide $1$ by $0.5$ .

$x = \frac{4}{- 2 \ln (1.9) + 1} - (2 \frac{\ln (5)}{- 2 \ln (1.9) + 1})$

Step 8.3.1.9

Combine $2$ and $\frac{\ln (5)}{- 2 \ln (1.9) + 1}$ .

$x = \frac{4}{- 2 \ln (1.9) + 1} - \frac{2 \ln (5)}{- 2 \ln (1.9) + 1}$

Step 8.3.2

Simplify terms.

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Step 8.3.2.1

Combine the numerators over the common denominator.

$x = \frac{4 - 2 \ln (5)}{- 2 \ln (1.9) + 1}$

Step 8.3.2.2

Factor $2$ out of $4 - 2 \ln (5)$ .

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Step 8.3.2.2.1

Factor $2$ out of $4$ .

$x = \frac{2 (2) - 2 \ln (5)}{- 2 \ln (1.9) + 1}$

Step 8.3.2.2.2

Factor $2$ out of $- 2 \ln (5)$ .

$x = \frac{2 (2) + 2 (- \ln (5))}{- 2 \ln (1.9) + 1}$

Step 8.3.2.2.3

Factor $2$ out of $2 (2) + 2 (- \ln (5))$ .

$x = \frac{2 (2 - \ln (5))}{- 2 \ln (1.9) + 1}$

Step 8.3.2.3

Factor $- 1$ out of $- 2 \ln (1.9)$ .

$x = \frac{2 (2 - \ln (5))}{- (2 \ln (1.9)) + 1}$

Step 8.3.2.4

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

$x = \frac{2 (2 - \ln (5))}{- (2 \ln (1.9)) - 1 (- 1)}$

Step 8.3.2.5

Factor $- 1$ out of $- (2 \ln (1.9)) - 1 (- 1)$ .

$x = \frac{2 (2 - \ln (5))}{- (2 \ln (1.9) - 1)}$

Step 8.3.2.6

Rewrite negatives.

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Step 8.3.2.6.1

Rewrite $- (2 \ln (1.9) - 1)$ as $- 1 (2 \ln (1.9) - 1)$ .

$x = \frac{2 (2 - \ln (5))}{- 1 (2 \ln (1.9) - 1)}$

Step 8.3.2.6.2

Move the negative in front of the fraction.

$x = - \frac{2 (2 - \ln (5))}{2 \ln (1.9) - 1}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2 (2 - \ln (5))}{2 \ln (1.9) - 1}$

Decimal Form:

$x = - 2.75327027 \dots$