Precalculus Examples

$\ln (2 x - 3) = 14$

Step 1

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

$e^{\ln (2 x - 3)} = e^{14}$

Step 2

Rewrite $\ln (2 x - 3) = 14$ 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 $b^{y} = x$ .

$e^{14} = 2 x - 3$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $2 x - 3 = e^{14}$ .

$2 x - 3 = e^{14}$

Step 3.2

Add $3$ to both sides of the equation.

$2 x = e^{14} + 3$

Step 3.3

Divide each term in $2 x = e^{14} + 3$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = e^{14} + 3$ by $2$ .

$\frac{2 x}{2} = \frac{e^{14}}{2} + \frac{3}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{e^{14}}{2} + \frac{3}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{14}}{2} + \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{14}}{2} + \frac{3}{2}$

Decimal Form:

$x = 601303.64208238 \dots$