Solve for n 5^(4n)>33

$5^{4 n} > 33$

Step 1

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

$\ln (5^{4 n}) > \ln (33)$

Step 2

Expand $\ln (5^{4 n})$ by moving $4 n$ outside the logarithm.

$4 n \ln (5) > \ln (33)$

Step 3

Divide each term in $4 n \ln (5) > \ln (33)$ by $4 \ln (5)$ and simplify.

Divide each term in $4 n \ln (5) > \ln (33)$ by $4 \ln (5)$ .

$\frac{4 n \ln (5)}{4 \ln (5)} > \frac{\ln (33)}{4 \ln (5)}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 n \ln (5)}{4 \ln (5)} > \frac{\ln (33)}{4 \ln (5)}$

Rewrite the expression.

$\frac{n \ln (5)}{\ln (5)} > \frac{\ln (33)}{4 \ln (5)}$

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

Cancel the common factor.

$\frac{n \ln (5)}{\ln (5)} > \frac{\ln (33)}{4 \ln (5)}$

Divide $n$ by $1$ .

$n > \frac{\ln (33)}{4 \ln (5)}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$n > \frac{\ln (33)}{4 \ln (5)}$

Interval Notation:

$(\frac{\ln (33)}{4 \ln (5)}, \infty)$

Step 5