Finite Math Examples

$e^{5 x + 2} = 0.4$

Step 1

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

$\ln (e^{5 x + 2}) = \ln (0.4)$

Step 2

Expand the left side.

Tap for more steps...

Step 2.1

Expand $\ln (e^{5 x + 2})$ by moving $5 x + 2$ outside the logarithm.

$(5 x + 2) \ln (e) = \ln (0.4)$

Step 2.2

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

$(5 x + 2) \cdot 1 = \ln (0.4)$

Step 2.3

Multiply $5 x + 2$ by $1$ .

$5 x + 2 = \ln (0.4)$

Step 3

Subtract $2$ from both sides of the equation.

$5 x = \ln (0.4) - 2$

Step 4

Divide each term in $5 x = \ln (0.4) - 2$ by $5$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $5 x = \ln (0.4) - 2$ by $5$ .

$\frac{5 x}{5} = \frac{\ln (0.4)}{5} + \frac{- 2}{5}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\ln (0.4)}{5} + \frac{- 2}{5}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (0.4)}{5} + \frac{- 2}{5}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Move the negative in front of the fraction.

$x = \frac{\ln (0.4)}{5} - \frac{2}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (0.4)}{5} - \frac{2}{5}$

Decimal Form:

$x = - 0.58325814 \dots$