Calculus Examples

$e^{6 x - 1} = 13$

Step 1

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

$\ln (e^{6 x - 1}) = \ln (13)$

Step 2

Expand the left side.

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Expand $\ln (e^{6 x - 1})$ by moving $6 x - 1$ outside the logarithm.

$(6 x - 1) \ln (e) = \ln (13)$

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

$(6 x - 1) \cdot 1 = \ln (13)$

Multiply $6 x - 1$ by $1$ .

$6 x - 1 = \ln (13)$

Step 3

Add $1$ to both sides of the equation.

$6 x = \ln (13) + 1$

Step 4

Divide each term in $6 x = \ln (13) + 1$ by $6$ and simplify.

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Divide each term in $6 x = \ln (13) + 1$ by $6$ .

$\frac{6 x}{6} = \frac{\ln (13)}{6} + \frac{1}{6}$

Simplify the left side.

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Cancel the common factor of $6$ .

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Cancel the common factor.

$\frac{6 x}{6} = \frac{\ln (13)}{6} + \frac{1}{6}$

Divide $x$ by $1$ .

$x = \frac{\ln (13)}{6} + \frac{1}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (13)}{6} + \frac{1}{6}$

Decimal Form:

$x = 0.59415822 \dots$