Precalculus Examples

$8^{y + 4} = 3$ , $e^{x} = 15$

Step 1

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

$\ln (8^{y + 4}) = \ln (3)$

Step 2

Expand $\ln (8^{y + 4})$ by moving $y + 4$ outside the logarithm.

$(y + 4) \ln (8) = \ln (3)$

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Apply the distributive property.

$y \ln (8) + 4 \ln (8) = \ln (3)$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$y \ln (8) + 4 \ln (8) - \ln (3) = 0$

Step 5

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 5.1

Subtract $4 \ln (8)$ from both sides of the equation.

$y \ln (8) - \ln (3) = - 4 \ln (8)$

Step 5.2

Add $\ln (3)$ to both sides of the equation.

$y \ln (8) = - 4 \ln (8) + \ln (3)$

Step 6

Divide each term in $y \ln (8) = - 4 \ln (8) + \ln (3)$ by $\ln (8)$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $y \ln (8) = - 4 \ln (8) + \ln (3)$ by $\ln (8)$ .

$\frac{y \ln (8)}{\ln (8)} = \frac{- 4 \ln (8)}{\ln (8)} + \frac{\ln (3)}{\ln (8)}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $\ln (8)$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{y \ln (8)}{\ln (8)} = \frac{- 4 \ln (8)}{\ln (8)} + \frac{\ln (3)}{\ln (8)}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4 \ln (8)}{\ln (8)} + \frac{\ln (3)}{\ln (8)}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Cancel the common factor of $\ln (8)$ .

Tap for more steps...

Step 6.3.1.1

Cancel the common factor.

$y = \frac{- 4 \ln (8)}{\ln (8)} + \frac{\ln (3)}{\ln (8)}$

Step 6.3.1.2

Divide $- 4$ by $1$ .

$y = - 4 + \frac{\ln (3)}{\ln (8)}$

Step 7

Replace all occurrences of $y$ with $- 4 + \frac{\ln (3)}{\ln (8)}$ in each equation.

Tap for more steps...

Step 7.1

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

$\ln (e^{x}) = \ln (15)$

$y = - 4 + \frac{\ln (3)}{\ln (8)}$

Step 7.2

Expand the left side.

Tap for more steps...

Step 7.2.1

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

$x \ln (e) = \ln (15)$

$y = - 4 + \frac{\ln (3)}{\ln (8)}$

Step 7.2.2

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

$x \cdot 1 = \ln (15)$

$y = - 4 + \frac{\ln (3)}{\ln (8)}$

Step 7.2.3

Multiply $x$ by $1$ .

$x = \ln (15)$

$y = - 4 + \frac{\ln (3)}{\ln (8)}$

$x = \ln (15)$

$y = - 4 + \frac{\ln (3)}{\ln (8)}$

$x = \ln (15)$

$y = - 4 + \frac{\ln (3)}{\ln (8)}$