Solve for y 3^(6y)>5^(y+2)

$3^{6 y} > 5^{y + 2}$

Step 1

Take the log of both sides of the inequality.

$\ln (3^{6 y}) > \ln (5^{y + 2})$

Step 2

Expand $\ln (3^{6 y})$ by moving $6 y$ outside the logarithm.

$6 y \ln (3) > \ln (5^{y + 2})$

Step 3

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

$6 y \ln (3) > (y + 2) \ln (5)$

Step 4

Solve the inequality for $y$ .

Apply the distributive property.

$6 y \ln (3) > y \ln (5) + 2 \ln (5)$

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

$6 y \ln (3) - y \ln (5) > 2 \ln (5)$

Factor $y$ out of $6 y \ln (3) - y \ln (5)$ .

Factor $y$ out of $6 y \ln (3)$ .

$y (6 \ln (3)) - y \ln (5) > 2 \ln (5)$

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

$y (6 \ln (3)) + y (- 1 \ln (5)) > 2 \ln (5)$

Factor $y$ out of $y (6 \ln (3)) + y (- 1 \ln (5))$ .

$y (6 \ln (3) - 1 \ln (5)) > 2 \ln (5)$

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

$y (6 \ln (3) - \ln (5)) > 2 \ln (5)$

Divide each term in $y (6 \ln (3) - \ln (5)) > 2 \ln (5)$ by $6 \ln (3) - \ln (5)$ and simplify.

Divide each term in $y (6 \ln (3) - \ln (5)) > 2 \ln (5)$ by $6 \ln (3) - \ln (5)$ .

$\frac{y (6 \ln (3) - \ln (5))}{6 \ln (3) - \ln (5)} > \frac{2 \ln (5)}{6 \ln (3) - \ln (5)}$

Simplify the left side.

Cancel the common factor of $6 \ln (3) - \ln (5)$ .

Cancel the common factor.

$\frac{y (6 \ln (3) - \ln (5))}{6 \ln (3) - \ln (5)} > \frac{2 \ln (5)}{6 \ln (3) - \ln (5)}$

Divide $y$ by $1$ .

$y > \frac{2 \ln (5)}{6 \ln (3) - \ln (5)}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$y > \frac{2 \ln (5)}{6 \ln (3) - \ln (5)}$

Interval Notation:

$(\frac{2 \ln (5)}{6 \ln (3) - \ln (5)}, \infty)$

Step 6