Precalculus Examples

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

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 2

Simplify the right side.

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Step 2.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Multiply $5$ by $- 2$ .

$\log (3 x) = \log (5 x - 10)$

Step 3

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$3 x = 5 x - 10$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $5 x$ from both sides of the equation.

$3 x - 5 x = - 10$

Step 4.1.2

Subtract $5 x$ from $3 x$ .

$- 2 x = - 10$

Step 4.2

Divide each term in $- 2 x = - 10$ by $- 2$ and simplify.

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Step 4.2.1

Divide each term in $- 2 x = - 10$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 10}{- 2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 10}{- 2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{- 2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 10$ by $- 2$ .

$x = 5$