Trigonometry Examples

$\log (2 x) - \log (3) = \log (x - 7)$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{2 x}{3}) = \log (x - 7)$

Step 2

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

$\frac{2 x}{3} = x - 7$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $3$ .

$\frac{2 x}{3} \cdot 3 = (x - 7) \cdot 3$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{2 x}{3} \cdot 3 = (x - 7) \cdot 3$

Step 3.2.1.1.2

Rewrite the expression.

$2 x = (x - 7) \cdot 3$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(x - 7) \cdot 3$ .

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Step 3.2.2.1.1

Apply the distributive property.

$2 x = x \cdot 3 - 7 \cdot 3$

Step 3.2.2.1.2

Simplify the expression.

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Step 3.2.2.1.2.1

Move $3$ to the left of $x$ .

$2 x = 3 \cdot x - 7 \cdot 3$

Step 3.2.2.1.2.2

Multiply $- 7$ by $3$ .

$2 x = 3 x - 21$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $3 x$ from both sides of the equation.

$2 x - 3 x = - 21$

Step 3.3.1.2

Subtract $3 x$ from $2 x$ .

$- x = - 21$

Step 3.3.2

Divide each term in $- x = - 21$ by $- 1$ and simplify.

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Step 3.3.2.1

Divide each term in $- x = - 21$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 21}{- 1}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 21}{- 1}$

Step 3.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 21}{- 1}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $- 21$ by $- 1$ .

$x = 21$