Precalculus Examples

$9 + 16 \ln (x) = 5$

Step 1

Move all terms not containing $\ln (x)$ to the right side of the equation.

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Subtract $9$ from both sides of the equation.

$16 \ln (x) = 5 - 9$

Subtract $9$ from $5$ .

$16 \ln (x) = - 4$

Step 2

Divide each term in $16 \ln (x) = - 4$ by $16$ and simplify.

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Divide each term in $16 \ln (x) = - 4$ by $16$ .

$\frac{16 \ln (x)}{16} = \frac{- 4}{16}$

Simplify the left side.

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Cancel the common factor of $16$ .

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Cancel the common factor.

$\frac{16 \ln (x)}{16} = \frac{- 4}{16}$

Divide $\ln (x)$ by $1$ .

$\ln (x) = \frac{- 4}{16}$

Simplify the right side.

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Cancel the common factor of $- 4$ and $16$ .

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Factor $4$ out of $- 4$ .

$\ln (x) = \frac{4 (- 1)}{16}$

Cancel the common factors.

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Factor $4$ out of $16$ .

$\ln (x) = \frac{4 \cdot - 1}{4 \cdot 4}$

Cancel the common factor.

$\ln (x) = \frac{4 \cdot - 1}{4 \cdot 4}$

Rewrite the expression.

$\ln (x) = \frac{- 1}{4}$

Move the negative in front of the fraction.

$\ln (x) = - \frac{1}{4}$

Step 3

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

$e^{\ln (x)} = e^{- \frac{1}{4}}$

Step 4

Rewrite $\ln (x) = - \frac{1}{4}$ 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 $b^{y} = x$ .

$e^{- \frac{1}{4}} = x$

Step 5

Solve for $x$ .

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Rewrite the equation as $x = e^{- \frac{1}{4}}$ .

$x = e^{- \frac{1}{4}}$

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

$x = \frac{1}{e^{\frac{1}{4}}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{e^{\frac{1}{4}}}$

Decimal Form:

$x = 0.77880078 \dots$