Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

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

$- 2 x + 3 = \frac{1}{e^{3}}$

Step 3.3

Subtract $3$ from both sides of the equation.

$- 2 x = \frac{1}{e^{3}} - 3$

Step 3.4

Divide each term in $- 2 x = \frac{1}{e^{3}} - 3$ by $- 2$ and simplify.

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Step 3.4.1

Divide each term in $- 2 x = \frac{1}{e^{3}} - 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{\frac{1}{e^{3}}}{- 2} + \frac{- 3}{- 2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{\frac{1}{e^{3}}}{- 2} + \frac{- 3}{- 2}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{1}{e^{3}}}{- 2} + \frac{- 3}{- 2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{e^{3}} \cdot \frac{1}{- 2} + \frac{- 3}{- 2}$

Step 3.4.3.1.2

Combine.

$x = \frac{1 \cdot 1}{e^{3} \cdot - 2} + \frac{- 3}{- 2}$

Step 3.4.3.1.3

Multiply $1$ by $1$ .

$x = \frac{1}{e^{3} \cdot - 2} + \frac{- 3}{- 2}$

Step 3.4.3.1.4

Move $- 2$ to the left of $e^{3}$ .

$x = \frac{1}{- 2 \cdot e^{3}} + \frac{- 3}{- 2}$

Step 3.4.3.1.5

Move the negative in front of the fraction.

$x = - \frac{1}{2 \cdot e^{3}} + \frac{- 3}{- 2}$

Step 3.4.3.1.6

Dividing two negative values results in a positive value.

$x = - \frac{1}{2 e^{3}} + \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2 e^{3}} + \frac{3}{2}$

Decimal Form:

$x = 1.47510646 \dots$