Calculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

Raise $x$ to the power of $1$ .

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

Step 1.3.2

Factor $x$ out of $x^{1}$ .

$\ln (\frac{x \cdot 1}{x^{2}} \cdot 6) = 0$

Step 1.3.3

Cancel the common factors.

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Step 1.3.3.1

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

$\ln (\frac{x \cdot 1}{x \cdot x} \cdot 6) = 0$

Step 1.3.3.2

Cancel the common factor.

$\ln (\frac{x \cdot 1}{x \cdot x} \cdot 6) = 0$

Step 1.3.3.3

Rewrite the expression.

$\ln (\frac{1}{x} \cdot 6) = 0$

Step 1.4

Combine $\frac{1}{x}$ and $6$ .

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

Step 2

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

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

Step 3

Rewrite $\ln (\frac{6}{x}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{6}{x}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{6}{x} = e^{0}$ .

$\frac{6}{x} = e^{0}$

Step 4.2

Anything raised to $0$ is $1$ .

$\frac{6}{x} = 1$

Step 4.3

Find the LCD of the terms in the equation.

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Step 4.3.1

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

$x, 1$

Step 4.3.2

The LCM of one and any expression is the expression.

$x$

Step 4.4

Multiply each term in $\frac{6}{x} = 1$ by $x$ to eliminate the fractions.

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Step 4.4.1

Multiply each term in $\frac{6}{x} = 1$ by $x$ .

$\frac{6}{x} x = 1 x$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $x$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{6}{x} x = 1 x$

Step 4.4.2.1.2

Rewrite the expression.

$6 = 1 x$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Multiply $x$ by $1$ .

$6 = x$

Step 4.5

Rewrite the equation as $x = 6$ .

$x = 6$