Precalculus Examples

$2 \log_{3} (x) - \frac{1}{2} \cdot (\log_{3} (x) + 5 \log_{3} (z))$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{3} (x)$ by moving $2$ inside the logarithm.

$\log_{3} (x^{2}) - \frac{1}{2} \cdot (\log_{3} (x) + 5 \log_{3} (z))$

Step 1.2

Simplify $5 \log_{3} (z)$ by moving $5$ inside the logarithm.

$\log_{3} (x^{2}) - \frac{1}{2} \cdot (\log_{3} (x) + \log_{3} (z^{5}))$

Step 1.3

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

$\log_{3} (x^{2}) - \frac{1}{2} \cdot \log_{3} (x z^{5})$

Step 1.4

Multiply $- \frac{1}{2} \log_{3} (x z^{5})$ .

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Step 1.4.1

Reorder $\log_{3} (x z^{5})$ and $\frac{1}{2}$ .

$\log_{3} (x^{2}) - (\frac{1}{2} \log_{3} (x z^{5}))$

Step 1.4.2

Simplify $\frac{1}{2} \log_{3} (x z^{5})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{3} (x^{2}) - \log_{3} ({(x z^{5})}^{\frac{1}{2}})$

Step 1.5

Apply the product rule to $x z^{5}$ .

$\log_{3} (x^{2}) - \log_{3} (x^{\frac{1}{2}} {(z^{5})}^{\frac{1}{2}})$

Step 1.6

Multiply the exponents in ${(z^{5})}^{\frac{1}{2}}$ .

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Step 1.6.1

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

$\log_{3} (x^{2}) - \log_{3} (x^{\frac{1}{2}} z^{5 (\frac{1}{2})})$

Step 1.6.2

Combine $5$ and $\frac{1}{2}$ .

$\log_{3} (x^{2}) - \log_{3} (x^{\frac{1}{2}} z^{\frac{5}{2}})$

Step 2

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

$\log_{3} (\frac{x^{2}}{x^{\frac{1}{2}} z^{\frac{5}{2}}})$

Step 3

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

$\log_{3} (\frac{x^{2} x^{- \frac{1}{2}}}{z^{\frac{5}{2}}})$

Step 4

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

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Step 4.1

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

$\log_{3} (\frac{x^{2 - \frac{1}{2}}}{z^{\frac{5}{2}}})$

Step 4.2

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

$\log_{3} (\frac{x^{2 \cdot \frac{2}{2} - \frac{1}{2}}}{z^{\frac{5}{2}}})$

Step 4.3

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

$\log_{3} (\frac{x^{\frac{2 \cdot 2}{2} - \frac{1}{2}}}{z^{\frac{5}{2}}})$

Step 4.4

Combine the numerators over the common denominator.

$\log_{3} (\frac{x^{\frac{2 \cdot 2 - 1}{2}}}{z^{\frac{5}{2}}})$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $2$ by $2$ .

$\log_{3} (\frac{x^{\frac{4 - 1}{2}}}{z^{\frac{5}{2}}})$

Step 4.5.2

Subtract $1$ from $4$ .

$\log_{3} (\frac{x^{\frac{3}{2}}}{z^{\frac{5}{2}}})$