Algebra Examples

$\log (\frac{10}{x}) = 2 - 2 \log (x)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log (\frac{10}{x}) + 2 \log (x) = 2$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\log (\frac{10}{x}) + 2 \log (x)$ .

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Step 2.1.1

Simplify $2 \log (x)$ by moving $2$ inside the logarithm.

$\log (\frac{10}{x}) + \log (x^{2}) = 2$

Step 2.1.2

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

$\log (\frac{10}{x} x^{2}) = 2$

Step 2.1.3

Cancel the common factor of $x$ .

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Step 2.1.3.1

Factor $x$ out of $x^{2}$ .

$\log (\frac{10}{x} (x \cdot x)) = 2$

Step 2.1.3.2

Cancel the common factor.

$\log (\frac{10}{x} (x \cdot x)) = 2$

Step 2.1.3.3

Rewrite the expression.

$\log (10 x) = 2$

Step 3

Rewrite $\log (10 x) = 2$ 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 $b^{y} = x$ .

$10^{2} = 10 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $10 x = 10^{2}$ .

$10 x = 10^{2}$

Step 4.2

Divide each term in $10 x = 10^{2}$ by $10$ and simplify.

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Step 4.2.1

Divide each term in $10 x = 10^{2}$ by $10$ .

$\frac{10 x}{10} = \frac{10^{2}}{10}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $10$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{10^{2}}{10}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10^{2}}{10}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $10^{2}$ and $10$ .

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Step 4.2.3.1.1

Factor $10$ out of $10^{2}$ .

$x = \frac{10 \cdot 10}{10}$

Step 4.2.3.1.2

Cancel the common factors.

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Step 4.2.3.1.2.1

Factor $10$ out of $10$ .

$x = \frac{10 \cdot 10}{10 (1)}$

Step 4.2.3.1.2.2

Cancel the common factor.

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

Step 4.2.3.1.2.3

Rewrite the expression.

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

Step 4.2.3.1.2.4

Divide $10$ by $1$ .

$x = 10$