Precalculus Examples

$e^{6 y} = 8$

Step 1

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

$\ln (e^{6 y}) = \ln (8)$

Step 2

Expand the left side.

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Step 2.1

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

$6 y \ln (e) = \ln (8)$

Step 2.2

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

$6 y \cdot 1 = \ln (8)$

Step 2.3

Multiply $6$ by $1$ .

$6 y = \ln (8)$

Step 3

Divide each term in $6 y = \ln (8)$ by $6$ and simplify.

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Step 3.1

Divide each term in $6 y = \ln (8)$ by $6$ .

$\frac{6 y}{6} = \frac{\ln (8)}{6}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $6$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{\ln (8)}{6}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (8)}{6}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$y = \frac{\ln (2^{3})}{6}$

Step 3.3.2

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

$y = \frac{3 \ln (2)}{6}$

Step 3.3.3

Cancel the common factor of $3$ and $6$ .

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Step 3.3.3.1

Factor $3$ out of $3 \ln (2)$ .

$y = \frac{3 (\ln (2))}{6}$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $3$ out of $6$ .

$y = \frac{3 \ln (2)}{3 \cdot 2}$

Step 3.3.3.2.2

Cancel the common factor.

$y = \frac{3 \ln (2)}{3 \cdot 2}$

Step 3.3.3.2.3

Rewrite the expression.

$y = \frac{\ln (2)}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\ln (2)}{2}$

Decimal Form:

$y = 0.34657359 \dots$