Solve for y 2^y=5^(y-2)

$2^{y} = 5^{y - 2}$

Step 1

Take the log of both sides of the equation.

$\ln (2^{y}) = \ln (5^{y - 2})$

Step 2

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

$y \ln (2) = \ln (5^{y - 2})$

Step 3

Expand $\ln (5^{y - 2})$ by moving $y - 2$ outside the logarithm.

$y \ln (2) = (y - 2) \ln (5)$

Step 4

Solve the equation for $y$ .

Apply the distributive property.

$y \ln (2) = y \ln (5) - 2 \ln (5)$

Subtract $y \ln (5)$ from both sides of the equation.

$y \ln (2) - y \ln (5) = - 2 \ln (5)$

Factor $y$ out of $y \ln (2) - y \ln (5)$ .

Factor $y$ out of $y \ln (2)$ .

$y (\ln (2)) - y \ln (5) = - 2 \ln (5)$

Factor $y$ out of $- y \ln (5)$ .

$y (\ln (2)) + y (- 1 \ln (5)) = - 2 \ln (5)$

Factor $y$ out of $y (\ln (2)) + y (- 1 \ln (5))$ .

$y (\ln (2) - 1 \ln (5)) = - 2 \ln (5)$

Rewrite $- 1 \ln (5)$ as $- \ln (5)$ .

$y (\ln (2) - \ln (5)) = - 2 \ln (5)$

Divide each term in $y (\ln (2) - \ln (5)) = - 2 \ln (5)$ by $\ln (2) - \ln (5)$ and simplify.

Divide each term in $y (\ln (2) - \ln (5)) = - 2 \ln (5)$ by $\ln (2) - \ln (5)$ .

$\frac{y (\ln (2) - \ln (5))}{\ln (2) - \ln (5)} = \frac{- 2 \ln (5)}{\ln (2) - \ln (5)}$

Simplify the left side.

Cancel the common factor of $\ln (2) - \ln (5)$ .

Cancel the common factor.

$\frac{y (\ln (2) - \ln (5))}{\ln (2) - \ln (5)} = \frac{- 2 \ln (5)}{\ln (2) - \ln (5)}$

Divide $y$ by $1$ .

$y = \frac{- 2 \ln (5)}{\ln (2) - \ln (5)}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{2 \ln (5)}{\ln (2) - \ln (5)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{2 \ln (5)}{\ln (2) - \ln (5)}$

Decimal Form:

$y = 3.51294159 \dots$