Solve for x 9^(3x-5)<2

$9^{3 x - 5} < 2$

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

$\ln (9^{3 x - 5}) < \ln (2)$

Expand $\ln (9^{3 x - 5})$ by moving $3 x - 5$ outside the logarithm.

$(3 x - 5) \ln (9) < \ln (2)$

Simplify the left side.

Apply the distributive property.

$3 x \ln (9) - 5 \ln (9) < \ln (2)$

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

$3 x \ln (9) - 5 \ln (9) - \ln (2) = 0$

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

Add $5 \ln (9)$ to both sides of the equation.

$3 x \ln (9) - \ln (2) = 5 \ln (9)$

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

$3 x \ln (9) = 5 \ln (9) + \ln (2)$

Divide each term in $3 x \ln (9) = 5 \ln (9) + \ln (2)$ by $3 \ln (9)$ and simplify.

Divide each term in $3 x \ln (9) = 5 \ln (9) + \ln (2)$ by $3 \ln (9)$ .

$\frac{3 x \ln (9)}{3 \ln (9)} = \frac{5 \ln (9)}{3 \ln (9)} + \frac{\ln (2)}{3 \ln (9)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x \ln (9)}{3 \ln (9)} = \frac{5 \ln (9)}{3 \ln (9)} + \frac{\ln (2)}{3 \ln (9)}$

Rewrite the expression.

$\frac{x \ln (9)}{\ln (9)} = \frac{5 \ln (9)}{3 \ln (9)} + \frac{\ln (2)}{3 \ln (9)}$

Cancel the common factor of $\ln (9)$ .

Cancel the common factor.

$\frac{x \ln (9)}{\ln (9)} = \frac{5 \ln (9)}{3 \ln (9)} + \frac{\ln (2)}{3 \ln (9)}$

Divide $x$ by $1$ .

$x = \frac{5 \ln (9)}{3 \ln (9)} + \frac{\ln (2)}{3 \ln (9)}$

Simplify the right side.

Cancel the common factor of $\ln (9)$ .

Cancel the common factor.

$x = \frac{5 \ln (9)}{3 \ln (9)} + \frac{\ln (2)}{3 \ln (9)}$

Rewrite the expression.

$x = \frac{5}{3} + \frac{\ln (2)}{3 \ln (9)}$

The solution consists of all of the true intervals.

$x < \frac{5}{3} + \frac{\ln (2)}{3 \ln (9)}$

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{5}{3} + \frac{\ln (2)}{3 \ln (9)}$

Interval Notation:

$(- \infty, \frac{5}{3} + \frac{\ln (2)}{3 \ln (9)})$