Solve for x 3^(x+8)=5

$3^{x + 8} = 5$

Step 1

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

$\ln (3^{x + 8}) = \ln (5)$

Step 2

Expand $\ln (3^{x + 8})$ by moving $x + 8$ outside the logarithm.

$(x + 8) \ln (3) = \ln (5)$

Step 3

Simplify the left side.

Apply the distributive property.

$x \ln (3) + 8 \ln (3) = \ln (5)$

Step 4

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

$x \ln (3) + 8 \ln (3) - \ln (5) = 0$

Step 5

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

Subtract $8 \ln (3)$ from both sides of the equation.

$x \ln (3) - \ln (5) = - 8 \ln (3)$

Add $\ln (5)$ to both sides of the equation.

$x \ln (3) = - 8 \ln (3) + \ln (5)$

Step 6

Divide each term in $x \ln (3) = - 8 \ln (3) + \ln (5)$ by $\ln (3)$ and simplify.

Divide each term in $x \ln (3) = - 8 \ln (3) + \ln (5)$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{- 8 \ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

Simplify the left side.

Cancel the common factor of $\ln (3)$ .

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{- 8 \ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

Divide $x$ by $1$ .

$x = \frac{- 8 \ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

Simplify the right side.

Cancel the common factor of $\ln (3)$ .

Cancel the common factor.

$x = \frac{- 8 \ln (3)}{\ln (3)} + \frac{\ln (5)}{\ln (3)}$

Divide $- 8$ by $1$ .

$x = - 8 + \frac{\ln (5)}{\ln (3)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 8 + \frac{\ln (5)}{\ln (3)}$

Decimal Form:

$x = - 6.53502647 \dots$