Precalculus Examples

$14 + 20 \ln (7 x) = 54$

Step 1

Move all terms not containing $\ln (7 x)$ to the right side of the equation.

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Step 1.1

Subtract $14$ from both sides of the equation.

$20 \ln (7 x) = 54 - 14$

Step 1.2

Subtract $14$ from $54$ .

$20 \ln (7 x) = 40$

Step 2

Divide each term in $20 \ln (7 x) = 40$ by $20$ and simplify.

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Step 2.1

Divide each term in $20 \ln (7 x) = 40$ by $20$ .

$\frac{20 \ln (7 x)}{20} = \frac{40}{20}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $20$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{20 \ln (7 x)}{20} = \frac{40}{20}$

Step 2.2.1.2

Divide $\ln (7 x)$ by $1$ .

$\ln (7 x) = \frac{40}{20}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $40$ by $20$ .

$\ln (7 x) = 2$

Step 3

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

$e^{\ln (7 x)} = e^{2}$

Step 4

Rewrite $\ln (7 x) = 2$ 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 $b^{y} = x$ .

$e^{2} = 7 x$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $7 x = e^{2}$ .

$7 x = e^{2}$

Step 5.2

Divide each term in $7 x = e^{2}$ by $7$ and simplify.

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Step 5.2.1

Divide each term in $7 x = e^{2}$ by $7$ .

$\frac{7 x}{7} = \frac{e^{2}}{7}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $7$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{e^{2}}{7}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{2}}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{2}}{7}$

Decimal Form:

$x = 1.05557944 \dots$