Precalculus Examples

$\log_{3} (2 x + 5) - 1 = \log_{3} (x)$

Step 1

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

$\log_{3} (2 x + 5) - \log_{3} (x) = 1$

Step 2

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

$\log_{3} (\frac{2 x + 5}{x}) = 1$

Step 3

Rewrite $\log_{3} (\frac{2 x + 5}{x}) = 1$ 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 $b^{y} = x$ .

$3^{1} = \frac{2 x + 5}{x}$

Step 4

Cross multiply to remove the fraction.

$2 x + 5 = 3 (x)$

Step 5

Multiply $3$ by $x$ .

$2 x + 5 = 3 x$

Step 6

Move all terms containing $x$ to the left side of the equation.

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Step 6.1

Subtract $3 x$ from both sides of the equation.

$2 x + 5 - 3 x = 0$

Step 6.2

Subtract $3 x$ from $2 x$ .

$- x + 5 = 0$

Step 7

Subtract $5$ from both sides of the equation.

$- x = - 5$

Step 8

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 8.1

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Step 8.2.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $- 5$ by $- 1$ .

$x = 5$