Solve for x 6^x=2^(x+2)

$6^{x} = 2^{x + 2}$

Step 1

Take the log of both sides of the equation.

$\ln (6^{x}) = \ln (2^{x + 2})$

Step 2

Expand $\ln (6^{x})$ by moving $x$ outside the logarithm.

$x \ln (6) = \ln (2^{x + 2})$

Step 3

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

$x \ln (6) = (x + 2) \ln (2)$

Step 4

Solve the equation for $x$ .

Apply the distributive property.

$x \ln (6) = x \ln (2) + 2 \ln (2)$

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

$x \ln (6) - x \ln (2) = 2 \ln (2)$

Factor $x$ out of $x \ln (6) - x \ln (2)$ .

Factor $x$ out of $x \ln (6)$ .

$x (\ln (6)) - x \ln (2) = 2 \ln (2)$

Factor $x$ out of $- x \ln (2)$ .

$x (\ln (6)) + x (- 1 \ln (2)) = 2 \ln (2)$

Factor $x$ out of $x (\ln (6)) + x (- 1 \ln (2))$ .

$x (\ln (6) - 1 \ln (2)) = 2 \ln (2)$

Rewrite $- 1 \ln (2)$ as $- \ln (2)$ .

$x (\ln (6) - \ln (2)) = 2 \ln (2)$

Divide each term in $x (\ln (6) - \ln (2)) = 2 \ln (2)$ by $\ln (6) - \ln (2)$ and simplify.

Divide each term in $x (\ln (6) - \ln (2)) = 2 \ln (2)$ by $\ln (6) - \ln (2)$ .

$\frac{x (\ln (6) - \ln (2))}{\ln (6) - \ln (2)} = \frac{2 \ln (2)}{\ln (6) - \ln (2)}$

Simplify the left side.

Cancel the common factor of $\ln (6) - \ln (2)$ .

Cancel the common factor.

$\frac{x (\ln (6) - \ln (2))}{\ln (6) - \ln (2)} = \frac{2 \ln (2)}{\ln (6) - \ln (2)}$

Divide $x$ by $1$ .

$x = \frac{2 \ln (2)}{\ln (6) - \ln (2)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \ln (2)}{\ln (6) - \ln (2)}$

Decimal Form:

$x = 1.26185950 \dots$