Algebra Examples

$4 \ln (- x) + 3 = 21$

Step 1

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

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Step 1.1

Subtract $3$ from both sides of the equation.

$4 \ln (- x) = 21 - 3$

Step 1.2

Subtract $3$ from $21$ .

$4 \ln (- x) = 18$

Step 2

Divide each term in $4 \ln (- x) = 18$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 \ln (- x) = 18$ by $4$ .

$\frac{4 \ln (- x)}{4} = \frac{18}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 \ln (- x)}{4} = \frac{18}{4}$

Step 2.2.1.2

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

$\ln (- x) = \frac{18}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $18$ and $4$ .

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Step 2.3.1.1

Factor $2$ out of $18$ .

$\ln (- x) = \frac{2 (9)}{4}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $4$ .

$\ln (- x) = \frac{2 \cdot 9}{2 \cdot 2}$

Step 2.3.1.2.2

Cancel the common factor.

$\ln (- x) = \frac{2 \cdot 9}{2 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

$\ln (- x) = \frac{9}{2}$

Step 3

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

$e^{\ln (- x)} = e^{\frac{9}{2}}$

Step 4

Rewrite $\ln (- x) = \frac{9}{2}$ 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 $b^{y} = x$ .

$e^{\frac{9}{2}} = - x$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $- x = e^{\frac{9}{2}}$ .

$- x = e^{\frac{9}{2}}$

Step 5.2

Divide each term in $- x = e^{\frac{9}{2}}$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- x = e^{\frac{9}{2}}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{e^{\frac{9}{2}}}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{e^{\frac{9}{2}}}{- 1}$

Step 5.2.2.2

Divide $x$ by $1$ .

$x = \frac{e^{\frac{9}{2}}}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative one from the denominator of $\frac{e^{\frac{9}{2}}}{- 1}$ .

$x = - 1 \cdot e^{\frac{9}{2}}$

Step 5.2.3.2

Rewrite $- 1 \cdot e^{\frac{9}{2}}$ as $- e^{\frac{9}{2}}$ .

$x = - e^{\frac{9}{2}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - e^{\frac{9}{2}}$

Decimal Form:

$x = - 90.01713130 \dots$