Solve for y 12^(y-2)=20

$12^{y - 2} = 20$

Step 1

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

$\ln (12^{y - 2}) = \ln (20)$

Step 2

Expand $\ln (12^{y - 2})$ by moving $y - 2$ outside the logarithm.

$(y - 2) \ln (12) = \ln (20)$

Step 3

Simplify the left side.

Apply the distributive property.

$y \ln (12) - 2 \ln (12) = \ln (20)$

Step 4

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

$y \ln (12) - 2 \ln (12) - \ln (20) = 0$

Step 5

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

Add $2 \ln (12)$ to both sides of the equation.

$y \ln (12) - \ln (20) = 2 \ln (12)$

Add $\ln (20)$ to both sides of the equation.

$y \ln (12) = 2 \ln (12) + \ln (20)$

Step 6

Divide each term in $y \ln (12) = 2 \ln (12) + \ln (20)$ by $\ln (12)$ and simplify.

Divide each term in $y \ln (12) = 2 \ln (12) + \ln (20)$ by $\ln (12)$ .

$\frac{y \ln (12)}{\ln (12)} = \frac{2 \ln (12)}{\ln (12)} + \frac{\ln (20)}{\ln (12)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{y \ln (12)}{\ln (12)} = \frac{2 \ln (12)}{\ln (12)} + \frac{\ln (20)}{\ln (12)}$

Divide $y$ by $1$ .

$y = \frac{2 \ln (12)}{\ln (12)} + \frac{\ln (20)}{\ln (12)}$

Simplify the right side.

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

Cancel the common factor.

$y = \frac{2 \ln (12)}{\ln (12)} + \frac{\ln (20)}{\ln (12)}$

Divide $2$ by $1$ .

$y = 2 + \frac{\ln (20)}{\ln (12)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 2 + \frac{\ln (20)}{\ln (12)}$

Decimal Form:

$y = 3.20557135 \dots$