Solve for x 7^(x+2)=410

$7^{x + 2} = 410$

Step 1

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

$\ln (7^{x + 2}) = \ln (410)$

Step 2

Expand $\ln (7^{x + 2})$ by moving $x + 2$ outside the logarithm.

$(x + 2) \ln (7) = \ln (410)$

Step 3

Simplify the left side.

Apply the distributive property.

$x \ln (7) + 2 \ln (7) = \ln (410)$

Step 4

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

$x \ln (7) + 2 \ln (7) - \ln (410) = 0$

Step 5

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

Subtract $2 \ln (7)$ from both sides of the equation.

$x \ln (7) - \ln (410) = - 2 \ln (7)$

Add $\ln (410)$ to both sides of the equation.

$x \ln (7) = - 2 \ln (7) + \ln (410)$

Step 6

Divide each term in $x \ln (7) = - 2 \ln (7) + \ln (410)$ by $\ln (7)$ and simplify.

Divide each term in $x \ln (7) = - 2 \ln (7) + \ln (410)$ by $\ln (7)$ .

$\frac{x \ln (7)}{\ln (7)} = \frac{- 2 \ln (7)}{\ln (7)} + \frac{\ln (410)}{\ln (7)}$

Simplify the left side.

Cancel the common factor of $\ln (7)$ .

Cancel the common factor.

$\frac{x \ln (7)}{\ln (7)} = \frac{- 2 \ln (7)}{\ln (7)} + \frac{\ln (410)}{\ln (7)}$

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (7)}{\ln (7)} + \frac{\ln (410)}{\ln (7)}$

Simplify the right side.

Cancel the common factor of $\ln (7)$ .

Cancel the common factor.

$x = \frac{- 2 \ln (7)}{\ln (7)} + \frac{\ln (410)}{\ln (7)}$

Divide $- 2$ by $1$ .

$x = - 2 + \frac{\ln (410)}{\ln (7)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 2 + \frac{\ln (410)}{\ln (7)}$

Decimal Form:

$x = 1.09169319 \dots$