Solve the Inequality for n 100^n=2

$100^{n} = 2$

Step 1

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

$\ln (100^{n}) = \ln (2)$

Step 2

Expand $\ln (100^{n})$ by moving $n$ outside the logarithm.

$n \ln (100) = \ln (2)$

Step 3

Divide each term in $n \ln (100) = \ln (2)$ by $\ln (100)$ and simplify.

Divide each term in $n \ln (100) = \ln (2)$ by $\ln (100)$ .

$\frac{n \ln (100)}{\ln (100)} = \frac{\ln (2)}{\ln (100)}$

Simplify the left side.

Cancel the common factor of $\ln (100)$ .

Cancel the common factor.

$\frac{n \ln (100)}{\ln (100)} = \frac{\ln (2)}{\ln (100)}$

Divide $n$ by $1$ .

$n = \frac{\ln (2)}{\ln (100)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$n = \frac{\ln (2)}{\ln (100)}$

Decimal Form:

$n = 0.15051499 \dots$