Algebra Examples

$3 \ln (x) = a + 3 \ln (b)$

Step 1

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

$3 \ln (x) - 3 \ln (b) = a$

Step 2

Simplify the left side.

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Step 2.1

Simplify $3 \ln (x) - 3 \ln (b)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$\ln (x^{3}) - 3 \ln (b) = a$

Step 2.1.1.2

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

$\ln (x^{3}) - \ln (b^{3}) = a$

Step 2.1.2

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

$\ln (\frac{x^{3}}{b^{3}}) = a$

Step 3

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

$e^{\ln (\frac{x^{3}}{b^{3}})} = e^{a}$

Step 4

Rewrite $\ln (\frac{x^{3}}{b^{3}}) = a$ 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 $b^{y} = x$ .

$e^{a} = \frac{x^{3}}{b^{3}}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $\frac{x^{3}}{b^{3}} = e^{a}$ .

$\frac{x^{3}}{b^{3}} = e^{a}$

Step 5.2

Multiply both sides by $b^{3}$ .

$\frac{x^{3}}{b^{3}} b^{3} = e^{a} b^{3}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $b^{3}$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{x^{3}}{b^{3}} b^{3} = e^{a} b^{3}$

Step 5.3.1.1.2

Rewrite the expression.

$x^{3} = e^{a} b^{3}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Reorder factors in $e^{a} b^{3}$ .

$x^{3} = b^{3} e^{a}$

Step 5.4

Solve for $x$ .

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Step 5.4.1

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

$x = \sqrt[3]{b^{3} e^{a}}$

Step 5.4.2

Pull terms out from under the radical.

$x = b \sqrt[3]{e^{a}}$