Solve for x 12^x=4^(x+1)

$12^{x} = 4^{x + 1}$

Step 1

Take the log of both sides of the equation.

$\ln (12^{x}) = \ln (4^{x + 1})$

Step 2

Expand $\ln (12^{x})$ by moving $x$ outside the logarithm.

$x \ln (12) = \ln (4^{x + 1})$

Step 3

Expand $\ln (4^{x + 1})$ by moving $x + 1$ outside the logarithm.

$x \ln (12) = (x + 1) \ln (4)$

Step 4

Solve the equation for $x$ .

Simplify $(x + 1) \ln (4)$ .

Apply the distributive property.

$x \ln (12) = x \ln (4) + 1 \ln (4)$

Multiply $\ln (4)$ by $1$ .

$x \ln (12) = x \ln (4) + \ln (4)$

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

$x \ln (12) - x \ln (4) = \ln (4)$

Factor $x$ out of $x \ln (12) - x \ln (4)$ .

Factor $x$ out of $x \ln (12)$ .

$x (\ln (12)) - x \ln (4) = \ln (4)$

Factor $x$ out of $- x \ln (4)$ .

$x (\ln (12)) + x (- 1 \ln (4)) = \ln (4)$

Factor $x$ out of $x (\ln (12)) + x (- 1 \ln (4))$ .

$x (\ln (12) - 1 \ln (4)) = \ln (4)$

Rewrite $- 1 \ln (4)$ as $- \ln (4)$ .

$x (\ln (12) - \ln (4)) = \ln (4)$

Divide each term in $x (\ln (12) - \ln (4)) = \ln (4)$ by $\ln (12) - \ln (4)$ and simplify.

Divide each term in $x (\ln (12) - \ln (4)) = \ln (4)$ by $\ln (12) - \ln (4)$ .

$\frac{x (\ln (12) - \ln (4))}{\ln (12) - \ln (4)} = \frac{\ln (4)}{\ln (12) - \ln (4)}$

Simplify the left side.

Cancel the common factor of $\ln (12) - \ln (4)$ .

Cancel the common factor.

$\frac{x (\ln (12) - \ln (4))}{\ln (12) - \ln (4)} = \frac{\ln (4)}{\ln (12) - \ln (4)}$

Divide $x$ by $1$ .

$x = \frac{\ln (4)}{\ln (12) - \ln (4)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (4)}{\ln (12) - \ln (4)}$

Decimal Form:

$x = 1.26185950 \dots$