Calculus Examples

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

Step 1

To solve for $x$ , rewrite the equation using properties of logarithms.

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

Step 2

Rewrite $\ln (x) = - \frac{1}{2}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- \frac{1}{2}} = x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x = e^{- \frac{1}{2}}$ .

$x = e^{- \frac{1}{2}}$

Step 3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{e^{\frac{1}{2}}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{e^{\frac{1}{2}}}$

Decimal Form:

$x = 0.60653065 \dots$