Precalculus Examples

$\log (50 x) = 2 + \log (2 x - 3)$

Step 1

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

$\log (50 x) - \log (2 x - 3) = 2$

Step 2

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

$\log (\frac{50 x}{2 x - 3}) = 2$

Step 3

Rewrite $\log (\frac{50 x}{2 x - 3}) = 2$ 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 $b^{y} = x$ .

$10^{2} = \frac{50 x}{2 x - 3}$

Step 4

Cross multiply to remove the fraction.

$50 x = 10^{2} (2 x - 3)$

Step 5

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

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Step 5.1

Raise $10$ to the power of $2$ .

$50 x = 100 (2 x - 3)$

Step 5.2

Apply the distributive property.

$50 x = 100 (2 x) + 100 \cdot - 3$

Step 5.3

Multiply.

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Step 5.3.1

Multiply $2$ by $100$ .

$50 x = 200 x + 100 \cdot - 3$

Step 5.3.2

Multiply $100$ by $- 3$ .

$50 x = 200 x - 300$

Step 6

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

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Step 6.1

Subtract $200 x$ from both sides of the equation.

$50 x - 200 x = - 300$

Step 6.2

Subtract $200 x$ from $50 x$ .

$- 150 x = - 300$

Step 7

Divide each term in $- 150 x = - 300$ by $- 150$ and simplify.

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Step 7.1

Divide each term in $- 150 x = - 300$ by $- 150$ .

$\frac{- 150 x}{- 150} = \frac{- 300}{- 150}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 150$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 150 x}{- 150} = \frac{- 300}{- 150}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 300}{- 150}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 300$ by $- 150$ .

$x = 2$