Finite Math Examples

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

Step 1

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

$\log (\frac{x + 3}{3}) = \log (10 x + 3)$

Step 2

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

$\frac{x + 3}{3} = 10 x + 3$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $3$ .

$\frac{x + 3}{3} \cdot 3 = (10 x + 3) \cdot 3$

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.

$\frac{x + 3}{3} \cdot 3 = (10 x + 3) \cdot 3$

Step 3.2.1.1.2

Rewrite the expression.

$x + 3 = (10 x + 3) \cdot 3$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(10 x + 3) \cdot 3$ .

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Step 3.2.2.1.1

Apply the distributive property.

$x + 3 = 10 x \cdot 3 + 3 \cdot 3$

Step 3.2.2.1.2

Multiply.

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Step 3.2.2.1.2.1

Multiply $3$ by $10$ .

$x + 3 = 30 x + 3 \cdot 3$

Step 3.2.2.1.2.2

Multiply $3$ by $3$ .

$x + 3 = 30 x + 9$

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 $30 x$ from both sides of the equation.

$x + 3 - 30 x = 9$

Step 3.3.1.2

Subtract $30 x$ from $x$ .

$- 29 x + 3 = 9$

Step 3.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.3.2.1

Subtract $3$ from both sides of the equation.

$- 29 x = 9 - 3$

Step 3.3.2.2

Subtract $3$ from $9$ .

$- 29 x = 6$

Step 3.3.3

Divide each term in $- 29 x = 6$ by $- 29$ and simplify.

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Step 3.3.3.1

Divide each term in $- 29 x = 6$ by $- 29$ .

$\frac{- 29 x}{- 29} = \frac{6}{- 29}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 29$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 29 x}{- 29} = \frac{6}{- 29}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{- 29}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{6}{29}$

Step 4