Precalculus Examples

$\log_{b} (216^{\frac{2}{3}}) + \log_{b} (49^{\frac{1}{2}}) - \log_{b} (49)$

Step 1

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

$\log_{b} (216^{\frac{2}{3}} \cdot 49^{\frac{1}{2}}) - \log_{b} (49)$

Step 2

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

$\log_{b} (\frac{216^{\frac{2}{3}} \cdot 49^{\frac{1}{2}}}{49})$

Step 3

Move $49^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\log_{b} (\frac{216^{\frac{2}{3}}}{49 \cdot 49^{- \frac{1}{2}}})$

Step 4

Multiply $49$ by $49^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.1

Multiply $49$ by $49^{- \frac{1}{2}}$ .

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Step 4.1.1

Raise $49$ to the power of $1$ .

$\log_{b} (\frac{216^{\frac{2}{3}}}{49^{1} \cdot 49^{- \frac{1}{2}}})$

Step 4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{b} (\frac{216^{\frac{2}{3}}}{49^{1 - \frac{1}{2}}})$

Step 4.2

Write $1$ as a fraction with a common denominator.

$\log_{b} (\frac{216^{\frac{2}{3}}}{49^{\frac{2}{2} - \frac{1}{2}}})$

Step 4.3

Combine the numerators over the common denominator.

$\log_{b} (\frac{216^{\frac{2}{3}}}{49^{\frac{2 - 1}{2}}})$

Step 4.4

Subtract $1$ from $2$ .

$\log_{b} (\frac{216^{\frac{2}{3}}}{49^{\frac{1}{2}}})$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $216$ as $6^{3}$ .

$\log_{b} (\frac{{(6^{3})}^{\frac{2}{3}}}{49^{\frac{1}{2}}})$

Step 5.2

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

$\log_{b} (\frac{6^{3 (\frac{2}{3})}}{49^{\frac{1}{2}}})$

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Cancel the common factor.

$\log_{b} (\frac{6^{3 (\frac{2}{3})}}{49^{\frac{1}{2}}})$

Step 5.3.2

Rewrite the expression.

$\log_{b} (\frac{6^{2}}{49^{\frac{1}{2}}})$

Step 5.4

Raise $6$ to the power of $2$ .

$\log_{b} (\frac{36}{49^{\frac{1}{2}}})$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite $49$ as $7^{2}$ .

$\log_{b} (\frac{36}{{(7^{2})}^{\frac{1}{2}}})$

Step 6.2

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

$\log_{b} (\frac{36}{7^{2 (\frac{1}{2})}})$

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Cancel the common factor.

$\log_{b} (\frac{36}{7^{2 (\frac{1}{2})}})$

Step 6.3.2

Rewrite the expression.

$\log_{b} (\frac{36}{7^{1}})$

Step 6.4

Evaluate the exponent.

$\log_{b} (\frac{36}{7})$