Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Cross multiply to remove the fraction.

$2 x - 1 = e^{2} (x + 3)$

Step 4

Simplify $e^{2} (x + 3)$ .

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 5

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

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

Step 6

Add $1$ to both sides of the equation.

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

Step 7

Factor $x$ out of $2 x - e^{2} x$ .

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Step 7.1

Factor $x$ out of $2 x$ .

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

Step 7.2

Factor $x$ out of $- e^{2} x$ .

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

Step 7.3

Factor $x$ out of $x \cdot 2 + x (- e^{2})$ .

$x (2 - e^{2}) = 3 e^{2} + 1$

Step 8

Divide each term in $x (2 - e^{2}) = 3 e^{2} + 1$ by $2 - e^{2}$ and simplify.

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Step 8.1

Divide each term in $x (2 - e^{2}) = 3 e^{2} + 1$ by $2 - e^{2}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $2 - e^{2}$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $x$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

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

Step 9

Exclude the solutions that do not make $\ln (2 x - 1) - \ln (x + 3) = 2$ true.

$No solution$