Solve for x (0.50)^(9900/x)=0.30

${(0.5)}^{\frac{9900}{x}} = 0.3$

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

$\ln ({0.5}^{\frac{9900}{x}}) = \ln (0.3)$

Expand the left side.

Expand $\ln ({0.5}^{\frac{9900}{x}})$ by moving $\frac{9900}{x}$ outside the logarithm.

$\frac{9900}{x} \ln (0.5) = \ln (0.3)$

Combine $\frac{9900}{x}$ and $\ln (0.5)$ .

$\frac{9900 \ln (0.5)}{x} = \ln (0.3)$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

The LCM of one and any expression is the expression.

$x$

Multiply each term in $\frac{9900 \ln (0.5)}{x} = \ln (0.3)$ by $x$ to eliminate the fractions.

Multiply each term in $\frac{9900 \ln (0.5)}{x} = \ln (0.3)$ by $x$ .

$\frac{9900 \ln (0.5)}{x} x = \ln (0.3) x$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{9900 \ln (0.5)}{x} x = \ln (0.3) x$

Rewrite the expression.

$9900 \ln (0.5) = \ln (0.3) x$

Solve the equation.

Rewrite the equation as $\ln (0.3) x = 9900 \ln (0.5)$ .

$\ln (0.3) x = 9900 \ln (0.5)$

Divide each term in $\ln (0.3) x = 9900 \ln (0.5)$ by $\ln (0.3)$ and simplify.

Divide each term in $\ln (0.3) x = 9900 \ln (0.5)$ by $\ln (0.3)$ .

$\frac{\ln (0.3) x}{\ln (0.3)} = \frac{9900 \ln (0.5)}{\ln (0.3)}$

Simplify the left side.

Cancel the common factor of $\ln (0.3)$ .

Cancel the common factor.

$\frac{\ln (0.3) x}{\ln (0.3)} = \frac{9900 \ln (0.5)}{\ln (0.3)}$

Divide $x$ by $1$ .

$x = \frac{9900 \ln (0.5)}{\ln (0.3)}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9900 \ln (0.5)}{\ln (0.3)}$

Decimal Form:

$x = 5699.59476068 \dots$