Precalculus Examples

$e^{3 x - 5} = 4$

Step 1

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

$\ln (e^{3 x - 5}) = \ln (4)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{3 x - 5})$ by moving $3 x - 5$ outside the logarithm.

$(3 x - 5) \ln (e) = \ln (4)$

Step 2.2

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

$(3 x - 5) \cdot 1 = \ln (4)$

Step 2.3

Multiply $3 x - 5$ by $1$ .

$3 x - 5 = \ln (4)$

Step 3

Add $5$ to both sides of the equation.

$3 x = \ln (4) + 5$

Step 4

Divide each term in $3 x = \ln (4) + 5$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 x = \ln (4) + 5$ by $3$ .

$\frac{3 x}{3} = \frac{\ln (4)}{3} + \frac{5}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\ln (4)}{3} + \frac{5}{3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (4)}{3} + \frac{5}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (4)}{3} + \frac{5}{3}$

Decimal Form:

$x = 2.12876478 \dots$