Precalculus Examples

$\frac{1}{2} \cdot \log_{b} (x) + 2 \log_{b} (y) - 3 \log_{b} (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{b} (x)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{b} (x^{\frac{1}{2}}) + 2 \log_{b} (y) - 3 \log_{b} (x)$

Step 1.2

Simplify $2 \log_{b} (y)$ by moving $2$ inside the logarithm.

$\log_{b} (x^{\frac{1}{2}}) + \log_{b} (y^{2}) - 3 \log_{b} (x)$

Step 1.3

Simplify $- 3 \log_{b} (x)$ by moving $3$ inside the logarithm.

$\log_{b} (x^{\frac{1}{2}}) + \log_{b} (y^{2}) - \log_{b} (x^{3})$

Step 2

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

$\log_{b} (x^{\frac{1}{2}} y^{2}) - \log_{b} (x^{3})$

Step 3

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

$\log_{b} (\frac{x^{\frac{1}{2}} y^{2}}{x^{3}})$

Step 4

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

$\log_{b} (\frac{y^{2}}{x^{3} x^{- \frac{1}{2}}})$

Step 5

Multiply $x^{3}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 5.1

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

$\log_{b} (\frac{y^{2}}{x^{3 - \frac{1}{2}}})$

Step 5.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log_{b} (\frac{y^{2}}{x^{3 \cdot \frac{2}{2} - \frac{1}{2}}})$

Step 5.3

Combine $3$ and $\frac{2}{2}$ .

$\log_{b} (\frac{y^{2}}{x^{\frac{3 \cdot 2}{2} - \frac{1}{2}}})$

Step 5.4

Combine the numerators over the common denominator.

$\log_{b} (\frac{y^{2}}{x^{\frac{3 \cdot 2 - 1}{2}}})$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $3$ by $2$ .

$\log_{b} (\frac{y^{2}}{x^{\frac{6 - 1}{2}}})$

Step 5.5.2

Subtract $1$ from $6$ .

$\log_{b} (\frac{y^{2}}{x^{\frac{5}{2}}})$