Calculus Examples

$\log (2 x) = \log (4) + \log (x - 1)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\log (4) + \log (x - 1)$ .

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Step 1.1.1

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

$\log (2 x) = \log (4 (x - 1))$

Step 1.1.2

Apply the distributive property.

$\log (2 x) = \log (4 x + 4 \cdot - 1)$

Step 1.1.3

Multiply $4$ by $- 1$ .

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

Step 2

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

$2 x = 4 x - 4$

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 $4 x$ from both sides of the equation.

$2 x - 4 x = - 4$

Step 3.1.2

Subtract $4 x$ from $2 x$ .

$- 2 x = - 4$

Step 3.2

Divide each term in $- 2 x = - 4$ by $- 2$ and simplify.

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Step 3.2.1

Divide each term in $- 2 x = - 4$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 4}{- 2}$

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.

$\frac{- 2 x}{- 2} = \frac{- 4}{- 2}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 4$ by $- 2$ .

$x = 2$