Solve for a 9^(6a)<31

$9^{6 a} < 31$

Step 1

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

$\ln (9^{6 a}) < \ln (31)$

Step 2

Expand $\ln (9^{6 a})$ by moving $6 a$ outside the logarithm.

$6 a \ln (9) < \ln (31)$

Step 3

Divide each term in $6 a \ln (9) < \ln (31)$ by $6 \ln (9)$ and simplify.

Divide each term in $6 a \ln (9) < \ln (31)$ by $6 \ln (9)$ .

$\frac{6 a \ln (9)}{6 \ln (9)} < \frac{\ln (31)}{6 \ln (9)}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 a \ln (9)}{6 \ln (9)} < \frac{\ln (31)}{6 \ln (9)}$

Rewrite the expression.

$\frac{a \ln (9)}{\ln (9)} < \frac{\ln (31)}{6 \ln (9)}$

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

Cancel the common factor.

$\frac{a \ln (9)}{\ln (9)} < \frac{\ln (31)}{6 \ln (9)}$

Divide $a$ by $1$ .

$a < \frac{\ln (31)}{6 \ln (9)}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$a < \frac{\ln (31)}{6 \ln (9)}$

Interval Notation:

$(- \infty, \frac{\ln (31)}{6 \ln (9)})$

Step 5