Precalculus Examples

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

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

Step 1

Simplify the right side.

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Step 1.1

Simplify

log (5) + log (x - 2)

$\log (5) + \log (x - 2)$ .

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Step 1.1.1

Use the product property of logarithms,

{log}_{b} (x) + {log}_{b} (y) = {log}_{b} (x y)

$\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

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

Step 1.1.2

Apply the distributive property.

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

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

Step 1.1.3

Multiply

5

$5$ by

- 2

$- 2$ .

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

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

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

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

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

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

Step 2

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

$3 x = 5 x - 10$

Step 3

Solve for

x

$x$ .

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Step 3.1

Move all terms containing

x

$x$ to the left side of the equation.

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Step 3.1.1

Subtract

5 x

$5 x$ from both sides of the equation.

3 x - 5 x = - 10

$3 x - 5 x = - 10$

Step 3.1.2

Subtract

5 x

$5 x$ from

3 x

$3 x$ .

- 2 x = - 10

$- 2 x = - 10$

- 2 x = - 10

$- 2 x = - 10$

Step 3.2

Divide each term in

- 2 x = - 10

$- 2 x = - 10$ by

- 2

$- 2$ and simplify.

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Step 3.2.1

Divide each term in

- 2 x = - 10

$- 2 x = - 10$ by

- 2

$- 2$ .

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of

- 2

$- 2$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide

$x$ by

$1$ .

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

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

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide

$- 10$ by

$- 2$ .

$x = 5$

$x = 5$

$x = 5$

$x = 5$

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

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