Algebra Examples

$\log (8) - \log (2 x) = - 1$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (\frac{8}{2 x}) = - 1$

Step 1.2

Cancel the common factor of $8$ and $2$ .

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Step 1.2.1

Factor $2$ out of $8$ .

$\log (\frac{2 \cdot 4}{2 x}) = - 1$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $2$ out of $2 x$ .

$\log (\frac{2 \cdot 4}{2 (x)}) = - 1$

Step 1.2.2.2

Cancel the common factor.

$\log (\frac{2 \cdot 4}{2 x}) = - 1$

Step 1.2.2.3

Rewrite the expression.

$\log (\frac{4}{x}) = - 1$

Step 2

Rewrite $\log (\frac{4}{x}) = - 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{- 1} = \frac{4}{x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{4}{x} = 10^{- 1}$ .

$\frac{4}{x} = 10^{- 1}$

Step 3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{x} = \frac{1}{10}$

Step 3.3

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$4 \cdot 10 = x \cdot 1$

Step 3.4

Solve the equation for $x$ .

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Step 3.4.1

Rewrite the equation as $x \cdot 1 = 4 \cdot 10$ .

$x \cdot 1 = 4 \cdot 10$

Step 3.4.2

Multiply $x$ by $1$ .

$x = 4 \cdot 10$

Step 3.4.3

Multiply $4$ by $10$ .

$x = 40$