Precalculus Examples

$\log (x + 3) - \log (2 x - 5) = 3$

Step 1

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

$\log (\frac{x + 3}{2 x - 5}) = 3$

Step 2

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

$10^{3} = \frac{x + 3}{2 x - 5}$

Step 3

Cross multiply to remove the fraction.

$x + 3 = 10^{3} (2 x - 5)$

Step 4

Simplify $10^{3} (2 x - 5)$ .

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Step 4.1

Raise $10$ to the power of $3$ .

$x + 3 = 1000 (2 x - 5)$

Step 4.2

Apply the distributive property.

$x + 3 = 1000 (2 x) + 1000 \cdot - 5$

Step 4.3

Multiply.

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Step 4.3.1

Multiply $2$ by $1000$ .

$x + 3 = 2000 x + 1000 \cdot - 5$

Step 4.3.2

Multiply $1000$ by $- 5$ .

$x + 3 = 2000 x - 5000$

Step 5

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

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Step 5.1

Subtract $2000 x$ from both sides of the equation.

$x + 3 - 2000 x = - 5000$

Step 5.2

Subtract $2000 x$ from $x$ .

$- 1999 x + 3 = - 5000$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Subtract $3$ from both sides of the equation.

$- 1999 x = - 5000 - 3$

Step 6.2

Subtract $3$ from $- 5000$ .

$- 1999 x = - 5003$

Step 7

Divide each term in $- 1999 x = - 5003$ by $- 1999$ and simplify.

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Step 7.1

Divide each term in $- 1999 x = - 5003$ by $- 1999$ .

$\frac{- 1999 x}{- 1999} = \frac{- 5003}{- 1999}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 1999$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 1999 x}{- 1999} = \frac{- 5003}{- 1999}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5003}{- 1999}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Dividing two negative values results in a positive value.

$x = \frac{5003}{1999}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5003}{1999}$

Decimal Form:

$x = 2.50275137 \dots$

Mixed Number Form:

$x = 2 \frac{1005}{1999}$