Algebra Examples

$\ln (\frac{9}{8}) = 2 \ln (3) - \ln (x)$

Step 1

Rewrite the equation as $2 \ln (3) - \ln (x) = \ln (\frac{9}{8})$ .

$2 \ln (3) - \ln (x) = \ln (\frac{9}{8})$

Step 2

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

$2 \ln (3) - \ln (x) - \ln (\frac{9}{8}) = 0$

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 \ln (3) - \ln (x) - \ln (\frac{9}{8})$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

$\ln (3^{2}) - \ln (x) - \ln (\frac{9}{8}) = 0$

Step 3.1.1.2

Raise $3$ to the power of $2$ .

$\ln (9) - \ln (x) - \ln (\frac{9}{8}) = 0$

Step 3.1.2

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

$\ln (\frac{9}{x}) - \ln (\frac{9}{8}) = 0$

Step 3.1.3

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

$\ln (\frac{\frac{9}{x}}{\frac{9}{8}}) = 0$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{9}{x} \cdot \frac{8}{9}) = 0$

Step 3.1.5

Cancel the common factor of $9$ .

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Step 3.1.5.1

Cancel the common factor.

$\ln (\frac{9}{x} \cdot \frac{8}{9}) = 0$

Step 3.1.5.2

Rewrite the expression.

$\ln (\frac{1}{x} \cdot 8) = 0$

Step 3.1.6

Combine $\frac{1}{x}$ and $8$ .

$\ln (\frac{8}{x}) = 0$

Step 4

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

$e^{\ln (\frac{8}{x})} = e^{0}$

Step 5

Rewrite $\ln (\frac{8}{x}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{8}{x}$

Step 6

Solve for $x$ .

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Step 6.1

Rewrite the equation as $\frac{8}{x} = e^{0}$ .

$\frac{8}{x} = e^{0}$

Step 6.2

Anything raised to $0$ is $1$ .

$\frac{8}{x} = 1$

Step 6.3

Find the LCD of the terms in the equation.

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Step 6.3.1

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

$x, 1$

Step 6.3.2

The LCM of one and any expression is the expression.

$x$

Step 6.4

Multiply each term in $\frac{8}{x} = 1$ by $x$ to eliminate the fractions.

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Step 6.4.1

Multiply each term in $\frac{8}{x} = 1$ by $x$ .

$\frac{8}{x} x = 1 x$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $x$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{8}{x} x = 1 x$

Step 6.4.2.1.2

Rewrite the expression.

$8 = 1 x$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Multiply $x$ by $1$ .

$8 = x$

Step 6.5

Rewrite the equation as $x = 8$ .

$x = 8$