Solve for x 2^x=1/( natural log of 2)

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

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

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

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

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

Expand the right side.

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

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

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

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

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

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

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.52876637 \dots$