Algebra Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

Move $x^{- \frac{1}{2}}$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.3

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

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 6.5

Add $- 1$ and $2$ .

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

Step 7

Reorder factors in $\log_{3} (y x^{\frac{1}{2}} z^{3})$ .

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