Solve for x 13^(x+2)=5

$13^{x + 2} = 5$

Step 1

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

$\ln (13^{x + 2}) = \ln (5)$

Step 2

Expand $\ln (13^{x + 2})$ by moving $x + 2$ outside the logarithm.

$(x + 2) \ln (13) = \ln (5)$

Step 3

Simplify the left side.

Apply the distributive property.

$x \ln (13) + 2 \ln (13) = \ln (5)$

Step 4

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

$x \ln (13) + 2 \ln (13) - \ln (5) = 0$

Step 5

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

Subtract $2 \ln (13)$ from both sides of the equation.

$x \ln (13) - \ln (5) = - 2 \ln (13)$

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

$x \ln (13) = - 2 \ln (13) + \ln (5)$

Step 6

Divide each term in $x \ln (13) = - 2 \ln (13) + \ln (5)$ by $\ln (13)$ and simplify.

Divide each term in $x \ln (13) = - 2 \ln (13) + \ln (5)$ by $\ln (13)$ .

$\frac{x \ln (13)}{\ln (13)} = \frac{- 2 \ln (13)}{\ln (13)} + \frac{\ln (5)}{\ln (13)}$

Simplify the left side.

Cancel the common factor of $\ln (13)$ .

Cancel the common factor.

$\frac{x \ln (13)}{\ln (13)} = \frac{- 2 \ln (13)}{\ln (13)} + \frac{\ln (5)}{\ln (13)}$

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (13)}{\ln (13)} + \frac{\ln (5)}{\ln (13)}$

Simplify the right side.

Cancel the common factor of $\ln (13)$ .

Cancel the common factor.

$x = \frac{- 2 \ln (13)}{\ln (13)} + \frac{\ln (5)}{\ln (13)}$

Divide $- 2$ by $1$ .

$x = - 2 + \frac{\ln (5)}{\ln (13)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 2 + \frac{\ln (5)}{\ln (13)}$

Decimal Form:

$x = - 1.37252643 \dots$