Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the right side.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $x$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.52876637 \dots$