Solve for x 4^(x-4)=3^(2x)

$4^{x - 4} = 3^{2 x}$

Step 1

Take the log of both sides of the equation.

$\ln (4^{x - 4}) = \ln (3^{2 x})$

Step 2

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

$(x - 4) \ln (4) = \ln (3^{2 x})$

Step 3

Expand $\ln (3^{2 x})$ by moving $2 x$ outside the logarithm.

$(x - 4) \ln (4) = 2 x \ln (3)$

Step 4

Solve the equation for $x$ .

Simplify $(x - 4) \ln (4)$ .

Rewrite.

$0 + 0 + (x - 4) \ln (4) = 2 x \ln (3)$

Simplify by adding zeros.

$(x - 4) \ln (4) = 2 x \ln (3)$

Apply the distributive property.

$x \ln (4) - 4 \ln (4) = 2 x \ln (3)$

Subtract $2 x \ln (3)$ from both sides of the equation.

$x \ln (4) - 4 \ln (4) - 2 x \ln (3) = 0$

Add $4 \ln (4)$ to both sides of the equation.

$x \ln (4) - 2 x \ln (3) = 4 \ln (4)$

Factor $x$ out of $x \ln (4) - 2 x \ln (3)$ .

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

$x (\ln (4)) - 2 x \ln (3) = 4 \ln (4)$

Factor $x$ out of $- 2 x \ln (3)$ .

$x (\ln (4)) + x (- 2 \ln (3)) = 4 \ln (4)$

Factor $x$ out of $x (\ln (4)) + x (- 2 \ln (3))$ .

$x (\ln (4) - 2 \ln (3)) = 4 \ln (4)$

Divide each term in $x (\ln (4) - 2 \ln (3)) = 4 \ln (4)$ by $\ln (4) - 2 \ln (3)$ and simplify.

Divide each term in $x (\ln (4) - 2 \ln (3)) = 4 \ln (4)$ by $\ln (4) - 2 \ln (3)$ .

$\frac{x (\ln (4) - 2 \ln (3))}{\ln (4) - 2 \ln (3)} = \frac{4 \ln (4)}{\ln (4) - 2 \ln (3)}$

Simplify the left side.

Cancel the common factor of $\ln (4) - 2 \ln (3)$ .

Cancel the common factor.

$\frac{x (\ln (4) - 2 \ln (3))}{\ln (4) - 2 \ln (3)} = \frac{4 \ln (4)}{\ln (4) - 2 \ln (3)}$

Divide $x$ by $1$ .

$x = \frac{4 \ln (4)}{\ln (4) - 2 \ln (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 \ln (4)}{\ln (4) - 2 \ln (3)}$

Decimal Form:

$x = - 6.83804516 \dots$