Calculus Examples

$\log (4 x) = \log (2) + \log (x - 3)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\log (2) + \log (x - 3)$ .

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Step 1.1.1

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

$\log (4 x) = \log (2 (x - 3))$

Step 1.1.2

Apply the distributive property.

$\log (4 x) = \log (2 x + 2 \cdot - 3)$

Step 1.1.3

Multiply $2$ by $- 3$ .

$\log (4 x) = \log (2 x - 6)$

Step 2

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

$4 x = 2 x - 6$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $2 x$ from both sides of the equation.

$4 x - 2 x = - 6$

Step 3.1.2

Subtract $2 x$ from $4 x$ .

$2 x = - 6$

Step 3.2

Divide each term in $2 x = - 6$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = - 6$ by $2$ .

$\frac{2 x}{2} = \frac{- 6}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 6}{2}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 6$ by $2$ .

$x = - 3$

Step 4

Exclude the solutions that do not make $\log (4 x) = \log (2) + \log (x - 3)$ true.

$No solution$