Calculus Examples

$\ln (4 x) - 3 \ln (x^{2}) = \ln (2)$

Step 1

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

$\ln (4 x) - 3 \ln (x^{2}) - \ln (2) = 0$

Step 2

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

$\ln (\frac{4 x}{2}) - 3 \ln (x^{2}) = 0$

Step 3

Cancel the common factor of $4$ and $2$ .

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Step 3.1

Factor $2$ out of $4 x$ .

$\ln (\frac{2 (2 x)}{2}) - 3 \ln (x^{2}) = 0$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $2$ .

$\ln (\frac{2 (2 x)}{2 (1)}) - 3 \ln (x^{2}) = 0$

Step 3.2.2

Cancel the common factor.

$\ln (\frac{2 (2 x)}{2 \cdot 1}) - 3 \ln (x^{2}) = 0$

Step 3.2.3

Rewrite the expression.

$\ln (\frac{2 x}{1}) - 3 \ln (x^{2}) = 0$

Step 3.2.4

Divide $2 x$ by $1$ .

$\ln (2 x) - 3 \ln (x^{2}) = 0$

Step 4

Simplify the left side.

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Step 4.1

Simplify $\ln (2 x) - 3 \ln (x^{2})$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Simplify $- 3 \ln (x^{2})$ by moving $3$ inside the logarithm.

$\ln (2 x) - \ln ({(x^{2})}^{3}) = 0$

Step 4.1.1.2

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 4.1.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (2 x) - \ln (x^{2 \cdot 3}) = 0$

Step 4.1.1.2.2

Multiply $2$ by $3$ .

$\ln (2 x) - \ln (x^{6}) = 0$

Step 4.1.2

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

$\ln (\frac{2 x}{x^{6}}) = 0$

Step 4.1.3

Cancel the common factor of $x$ and $x^{6}$ .

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Step 4.1.3.1

Factor $x$ out of $2 x$ .

$\ln (\frac{x \cdot 2}{x^{6}}) = 0$

Step 4.1.3.2

Cancel the common factors.

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Step 4.1.3.2.1

Factor $x$ out of $x^{6}$ .

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

Step 4.1.3.2.2

Cancel the common factor.

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

Step 4.1.3.2.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Rewrite $\ln (\frac{2}{x^{5}}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{2}{x^{5}}$

Step 7

Solve for $x$ .

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Step 7.1

Rewrite the equation as $\frac{2}{x^{5}} = e^{0}$ .

$\frac{2}{x^{5}} = e^{0}$

Step 7.2

Anything raised to $0$ is $1$ .

$\frac{2}{x^{5}} = 1$

Step 7.3

Find the LCD of the terms in the equation.

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Step 7.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{5}, 1$

Step 7.3.2

The LCM of one and any expression is the expression.

$x^{5}$

Step 7.4

Multiply each term in $\frac{2}{x^{5}} = 1$ by $x^{5}$ to eliminate the fractions.

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Step 7.4.1

Multiply each term in $\frac{2}{x^{5}} = 1$ by $x^{5}$ .

$\frac{2}{x^{5}} x^{5} = 1 x^{5}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $x^{5}$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{2}{x^{5}} x^{5} = 1 x^{5}$

Step 7.4.2.1.2

Rewrite the expression.

$2 = 1 x^{5}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Multiply $x^{5}$ by $1$ .

$2 = x^{5}$

Step 7.5

Solve the equation.

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Step 7.5.1

Rewrite the equation as $x^{5} = 2$ .

$x^{5} = 2$

Step 7.5.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[5]{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt[5]{2}$

Decimal Form:

$x = 1.14869835 \dots$