Precalculus Examples

$\log (5 x) = \log (4) + \log (x - 2)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\log (4) + \log (x - 2)$ .

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Step 1.1.1

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

$\log (5 x) = \log (4 (x - 2))$

Step 1.1.2

Apply the distributive property.

$\log (5 x) = \log (4 x + 4 \cdot - 2)$

Step 1.1.3

Multiply $4$ by $- 2$ .

$\log (5 x) = \log (4 x - 8)$

Step 2

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

$5 x = 4 x - 8$

Step 3

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

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Step 3.1

Subtract $4 x$ from both sides of the equation.

$5 x - 4 x = - 8$

Step 3.2

Subtract $4 x$ from $5 x$ .

$x = - 8$

Step 4

Exclude the solutions that do not make $\log (5 x) = \log (4) + \log (x - 2)$ true.

$No solution$