Precalculus Examples

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

Step 1

Divide each term in $5 e^{x} = \ln (4)$ by $5$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $5 e^{x} = \ln (4)$ by $5$ .

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

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Divide $e^{x}$ by $1$ .

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

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Rewrite $\frac{\ln (4)}{5}$ as $\frac{1}{5} \ln (4)$ .

$e^{x} = \frac{1}{5} \ln (4)$

Step 1.3.2

Simplify $\frac{1}{5} \ln (4)$ by moving $\frac{1}{5}$ inside the logarithm.

$e^{x} = \ln (4^{\frac{1}{5}})$

Step 2

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

$\ln (e^{x}) = \ln (\ln (4^{\frac{1}{5}}))$

Step 3

Expand the left side.

Tap for more steps...

Step 3.1

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

$x \ln (e) = \ln (\ln (4^{\frac{1}{5}}))$

Step 3.2

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

$x \cdot 1 = \ln (\ln (4^{\frac{1}{5}}))$

Step 3.3

Multiply $x$ by $1$ .

$x = \ln (\ln (4^{\frac{1}{5}}))$

Step 4

Expand the right side.

Tap for more steps...

Step 4.1

Expand $\ln (4^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$x = \ln (\frac{1}{5} \ln (4))$

Step 4.2

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

Step 4.3

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

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

Step 4.4

Multiply $\frac{1}{5} (2 \ln (2))$ .

Tap for more steps...

Step 4.4.1

Combine $2$ and $\frac{1}{5}$ .

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

Step 4.4.2

Combine $\frac{2}{5}$ and $\ln (2)$ .

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

Step 4.5

Rewrite $\ln (\frac{2 \ln (2)}{5})$ as $\ln (2 \ln (2)) - \ln (5)$ .

$x = \ln (2 \ln (2)) - \ln (5)$

Step 4.6

Rewrite $\ln (2 \ln (2))$ as $\ln (2) + \ln (\ln (2))$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$x = \ln (2 \ln (2)) - \ln (5)$

Step 5.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 5.3

Simplify $2 \ln (2)$ by moving $2$ inside the logarithm.

$x = \ln (\frac{\ln (2^{2})}{5})$

Step 5.4

Raise $2$ to the power of $2$ .

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

Step 5.5

Rewrite $\frac{\ln (4)}{5}$ as $\frac{1}{5} \ln (4)$ .

$x = \ln (\frac{1}{5} \ln (4))$

Step 5.6

Simplify $\frac{1}{5} \ln (4)$ by moving $\frac{1}{5}$ inside the logarithm.

$x = \ln (\ln (4^{\frac{1}{5}}))$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \ln (\ln (4^{\frac{1}{5}}))$

Decimal Form:

$x = - 1.28280365 \dots$