Algebra Examples

$2 \ln (e^{\ln (2 x)}) - \ln (e^{\ln (10 x)}) = \ln (30)$

Step 1

Simplify each term.

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Step 1.1

Use logarithm rules to move $\ln (2 x)$ out of the exponent.

$2 (\ln (2 x) \ln (e)) - \ln (e^{\ln (10 x)}) = \ln (30)$

Step 1.2

The natural logarithm of $e$ is $1$ .

$2 (\ln (2 x) \cdot 1) - \ln (e^{\ln (10 x)}) = \ln (30)$

Step 1.3

Multiply $\ln (2 x)$ by $1$ .

$2 \ln (2 x) - \ln (e^{\ln (10 x)}) = \ln (30)$

Step 1.4

Use logarithm rules to move $\ln (10 x)$ out of the exponent.

$2 \ln (2 x) - (\ln (10 x) \ln (e)) = \ln (30)$

Step 1.5

The natural logarithm of $e$ is $1$ .

$2 \ln (2 x) - (\ln (10 x) \cdot 1) = \ln (30)$

Step 1.6

Multiply $\ln (10 x)$ by $1$ .

$2 \ln (2 x) - \ln (10 x) = \ln (30)$

Step 2

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

$2 \ln (2 x) - \ln (10 x) - \ln (30) = 0$

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 \ln (2 x) - \ln (10 x) - \ln (30)$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

$\ln ({(2 x)}^{2}) - \ln (10 x) - \ln (30) = 0$

Step 3.1.1.2

Apply the product rule to $2 x$ .

$\ln (2^{2} x^{2}) - \ln (10 x) - \ln (30) = 0$

Step 3.1.1.3

Raise $2$ to the power of $2$ .

$\ln (4 x^{2}) - \ln (10 x) - \ln (30) = 0$

Step 3.1.2

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

$\ln (\frac{4 x^{2}}{10 x}) - \ln (30) = 0$

Step 3.1.3

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

$\ln (\frac{\frac{4 x^{2}}{10 x}}{30}) = 0$

Step 3.1.4

Cancel the common factor of $4$ and $10$ .

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Step 3.1.4.1

Factor $2$ out of $4 x^{2}$ .

$\ln (\frac{\frac{2 (2 x^{2})}{10 x}}{30}) = 0$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $10 x$ .

$\ln (\frac{\frac{2 (2 x^{2})}{2 (5 x)}}{30}) = 0$

Step 3.1.4.2.2

Cancel the common factor.

$\ln (\frac{\frac{2 (2 x^{2})}{2 (5 x)}}{30}) = 0$

Step 3.1.4.2.3

Rewrite the expression.

$\ln (\frac{\frac{2 x^{2}}{5 x}}{30}) = 0$

Step 3.1.5

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

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Step 3.1.5.1

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

$\ln (\frac{\frac{x (2 x)}{5 x}}{30}) = 0$

Step 3.1.5.2

Cancel the common factors.

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Step 3.1.5.2.1

Factor $x$ out of $5 x$ .

$\ln (\frac{\frac{x (2 x)}{x \cdot 5}}{30}) = 0$

Step 3.1.5.2.2

Cancel the common factor.

$\ln (\frac{\frac{x (2 x)}{x \cdot 5}}{30}) = 0$

Step 3.1.5.2.3

Rewrite the expression.

$\ln (\frac{\frac{2 x}{5}}{30}) = 0$

Step 3.1.6

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{2 x}{5} \cdot \frac{1}{30}) = 0$

Step 3.1.7

Cancel the common factor of $2$ .

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Step 3.1.7.1

Factor $2$ out of $2 x$ .

$\ln (\frac{2 (x)}{5} \cdot \frac{1}{30}) = 0$

Step 3.1.7.2

Factor $2$ out of $30$ .

$\ln (\frac{2 (x)}{5} \cdot \frac{1}{2 (15)}) = 0$

Step 3.1.7.3

Cancel the common factor.

$\ln (\frac{2 x}{5} \cdot \frac{1}{2 \cdot 15}) = 0$

Step 3.1.7.4

Rewrite the expression.

$\ln (\frac{x}{5} \cdot \frac{1}{15}) = 0$

Step 3.1.8

Multiply $\frac{x}{5}$ by $\frac{1}{15}$ .

$\ln (\frac{x}{5 \cdot 15}) = 0$

Step 3.1.9

Multiply $5$ by $15$ .

$\ln (\frac{x}{75}) = 0$

Step 4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{x}{75})} = e^{0}$

Step 5

Rewrite $\ln (\frac{x}{75}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{x}{75}$

Step 6

Solve for $x$ .

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Step 6.1

Rewrite the equation as $\frac{x}{75} = e^{0}$ .

$\frac{x}{75} = e^{0}$

Step 6.2

Multiply both sides of the equation by $75$ .

$75 \frac{x}{75} = 75 e^{0}$

Step 6.3

Simplify both sides of the equation.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Cancel the common factor of $75$ .

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Step 6.3.1.1.1

Cancel the common factor.

$75 \frac{x}{75} = 75 e^{0}$

Step 6.3.1.1.2

Rewrite the expression.

$x = 75 e^{0}$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Simplify $75 e^{0}$ .

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Step 6.3.2.1.1

Anything raised to $0$ is $1$ .

$x = 75 \cdot 1$

Step 6.3.2.1.2

Multiply $75$ by $1$ .

$x = 75$