Solve for x (1/2)^x=1

${(\frac{1}{2})}^{x} = 1$

Step 1

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

$\ln ({(\frac{1}{2})}^{x}) = \ln (1)$

Step 2

Expand the left side.

Expand $\ln ({(\frac{1}{2})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{1}{2}) = \ln (1)$

Rewrite $\ln (\frac{1}{2})$ as $\ln (1) - \ln (2)$ .

$x (\ln (1) - \ln (2)) = \ln (1)$

The natural logarithm of $1$ is $0$ .

$x (0 - \ln (2)) = \ln (1)$

Subtract $\ln (2)$ from $0$ .

$x (- \ln (2)) = \ln (1)$

Step 3

Simplify the left side.

Reorder factors in $x (- \ln (2))$ .

$- x \ln (2) = \ln (1)$

Step 4

Simplify the right side.

The natural logarithm of $1$ is $0$ .

$- x \ln (2) = 0$

Step 5

Divide each term in $- x \ln (2) = 0$ by $- \ln (2)$ and simplify.

Divide each term in $- x \ln (2) = 0$ by $- \ln (2)$ .

$\frac{- x \ln (2)}{- \ln (2)} = \frac{0}{- \ln (2)}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x \ln (2)}{\ln (2)} = \frac{0}{- \ln (2)}$

Cancel the common factor of $\ln (2)$ .

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{0}{- \ln (2)}$

Divide $x$ by $1$ .

$x = \frac{0}{- \ln (2)}$

Simplify the right side.

Divide $0$ by $- \ln (2)$ .

$x = 0$