Algebra Examples

$\ln (3 x + 2) = 1 + \ln (2 x)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\ln (3 x + 2) - \ln (2 x) = 1$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{3 x + 2}{2 x}) = 1$

Step 3

Rewrite $\ln (\frac{3 x + 2}{2 x}) = 1$ 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 $b^{y} = x$ .

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

Step 4

Cross multiply to remove the fraction.

$3 x + 2 = e (2 x)$

Step 5

Move $2$ to the left of $e$ .

$3 x + 2 = 2 e x$

Step 6

Subtract $2 e x$ from both sides of the equation.

$3 x + 2 - 2 e x = 0$

Step 7

Subtract $2$ from both sides of the equation.

$3 x - 2 e x = - 2$

Step 8

Factor $x$ out of $3 x - 2 e x$ .

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Step 8.1

Factor $x$ out of $3 x$ .

$x \cdot 3 - 2 e x = - 2$

Step 8.2

Factor $x$ out of $- 2 e x$ .

$x \cdot 3 + x (- 2 e) = - 2$

Step 8.3

Factor $x$ out of $x \cdot 3 + x (- 2 e)$ .

$x (3 - 2 e) = - 2$

Step 9

Divide each term in $x (3 - 2 e) = - 2$ by $3 - 2 e$ and simplify.

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Step 9.1

Divide each term in $x (3 - 2 e) = - 2$ by $3 - 2 e$ .

$\frac{x (3 - 2 e)}{3 - 2 e} = \frac{- 2}{3 - 2 e}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $3 - 2 e$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{x (3 - 2 e)}{3 - 2 e} = \frac{- 2}{3 - 2 e}$

Step 9.2.1.2

Divide $x$ by $1$ .

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

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.82082813 \dots$