Algebra Examples

$\sqrt{648} = \sqrt{2^{a} \cdot 3^{b}}$

Step 1

Rewrite the equation as $\sqrt{2^{a} \cdot 3^{b}} = \sqrt{648}$ .

$\sqrt{2^{a} \cdot 3^{b}} = \sqrt{648}$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (\sqrt{2^{a} \cdot 3^{b}}) = \ln (\sqrt{648})$

Step 3

Expand the left side.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2^{a} \cdot 3^{b}}$ as ${(2^{a} \cdot 3^{b})}^{\frac{1}{2}}$ .

$\ln ({(2^{a} \cdot 3^{b})}^{\frac{1}{2}}) = \ln (\sqrt{648})$

Step 3.2

Expand $\ln ({(2^{a} \cdot 3^{b})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (2^{a} \cdot 3^{b}) = \ln (\sqrt{648})$

Step 3.3

Rewrite $\ln (2^{a} \cdot 3^{b})$ as $\ln (2^{a}) + \ln (3^{b})$ .

$\frac{1}{2} (\ln (2^{a}) + \ln (3^{b})) = \ln (\sqrt{648})$

Step 3.4

Expand $\ln (2^{a})$ by moving $a$ outside the logarithm.

$\frac{1}{2} (a \ln (2) + \ln (3^{b})) = \ln (\sqrt{648})$

Step 3.5

Expand $\ln (3^{b})$ by moving $b$ outside the logarithm.

$\frac{1}{2} (a \ln (2) + b \ln (3)) = \ln (\sqrt{648})$

Step 4

Simplify the left side.

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Step 4.1

Simplify $\frac{1}{2} (a \ln (2) + b \ln (3))$ .

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Step 4.1.1

Apply the distributive property.

$\frac{1}{2} (a \ln (2)) + \frac{1}{2} (b \ln (3)) = \ln (\sqrt{648})$

Step 4.1.2

Multiply $\frac{1}{2} (a \ln (2))$ .

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Step 4.1.2.1

Combine $a$ and $\frac{1}{2}$ .

$\frac{a}{2} \ln (2) + \frac{1}{2} (b \ln (3)) = \ln (\sqrt{648})$

Step 4.1.2.2

Combine $\frac{a}{2}$ and $\ln (2)$ .

$\frac{a \ln (2)}{2} + \frac{1}{2} (b \ln (3)) = \ln (\sqrt{648})$

Step 4.1.3

Multiply $\frac{1}{2} (b \ln (3))$ .

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Step 4.1.3.1

Combine $b$ and $\frac{1}{2}$ .

$\frac{a \ln (2)}{2} + \frac{b}{2} \ln (3) = \ln (\sqrt{648})$

Step 4.1.3.2

Combine $\frac{b}{2}$ and $\ln (3)$ .

$\frac{a \ln (2)}{2} + \frac{b \ln (3)}{2} = \ln (\sqrt{648})$

Step 5

Simplify the right side.

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Step 5.1

Simplify $\ln (\sqrt{648})$ .

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Step 5.1.1

Rewrite $648$ as $18^{2} \cdot 2$ .

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Step 5.1.1.1

Factor $324$ out of $648$ .

$\frac{a \ln (2)}{2} + \frac{b \ln (3)}{2} = \ln (\sqrt{324 (2)})$

Step 5.1.1.2

Rewrite $324$ as $18^{2}$ .

$\frac{a \ln (2)}{2} + \frac{b \ln (3)}{2} = \ln (\sqrt{18^{2} \cdot 2})$

Step 5.1.2

Pull terms out from under the radical.

$\frac{a \ln (2)}{2} + \frac{b \ln (3)}{2} = \ln (18 \sqrt{2})$

Step 6

Multiply each term in $\frac{a \ln (2)}{2} + \frac{b \ln (3)}{2} = \ln (18 \sqrt{2})$ by $2$ to eliminate the fractions.

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Step 6.1

Multiply each term in $\frac{a \ln (2)}{2} + \frac{b \ln (3)}{2} = \ln (18 \sqrt{2})$ by $2$ .

$\frac{a \ln (2)}{2} \cdot 2 + \frac{b \ln (3)}{2} \cdot 2 = \ln (18 \sqrt{2}) \cdot 2$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1

Cancel the common factor.

$\frac{a \ln (2)}{2} \cdot 2 + \frac{b \ln (3)}{2} \cdot 2 = \ln (18 \sqrt{2}) \cdot 2$

Step 6.2.1.1.2

Rewrite the expression.

$a \ln (2) + \frac{b \ln (3)}{2} \cdot 2 = \ln (18 \sqrt{2}) \cdot 2$

Step 6.2.1.2

Cancel the common factor of $2$ .

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Step 6.2.1.2.1

Cancel the common factor.

$a \ln (2) + \frac{b \ln (3)}{2} \cdot 2 = \ln (18 \sqrt{2}) \cdot 2$

Step 6.2.1.2.2

Rewrite the expression.

$a \ln (2) + b \ln (3) = \ln (18 \sqrt{2}) \cdot 2$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move $2$ to the left of $\ln (18 \sqrt{2})$ .

$a \ln (2) + b \ln (3) = 2 \ln (18 \sqrt{2})$

Step 7

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

$a \ln (2) + b \ln (3) - 2 \ln (18 \sqrt{2}) = 0$

Step 8

Move all terms not containing $a$ to the right side of the equation.

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Step 8.1

Subtract $b \ln (3)$ from both sides of the equation.

$a \ln (2) - 2 \ln (18 \sqrt{2}) = - b \ln (3)$

Step 8.2

Add $2 \ln (18 \sqrt{2})$ to both sides of the equation.

$a \ln (2) = - b \ln (3) + 2 \ln (18 \sqrt{2})$

Step 9

Divide each term in $a \ln (2) = - b \ln (3) + 2 \ln (18 \sqrt{2})$ by $\ln (2)$ and simplify.

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Step 9.1

Divide each term in $a \ln (2) = - b \ln (3) + 2 \ln (18 \sqrt{2})$ by $\ln (2)$ .

$\frac{a \ln (2)}{\ln (2)} = \frac{- b \ln (3)}{\ln (2)} + \frac{2 \ln (18 \sqrt{2})}{\ln (2)}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $\ln (2)$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{a \ln (2)}{\ln (2)} = \frac{- b \ln (3)}{\ln (2)} + \frac{2 \ln (18 \sqrt{2})}{\ln (2)}$

Step 9.2.1.2

Divide $a$ by $1$ .

$a = \frac{- b \ln (3)}{\ln (2)} + \frac{2 \ln (18 \sqrt{2})}{\ln (2)}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Move the negative in front of the fraction.

$a = - \frac{b \ln (3)}{\ln (2)} + \frac{2 \ln (18 \sqrt{2})}{\ln (2)}$