Precalculus Examples

$0.5 \log_{x} (y) + 5 \log_{x} (z) - 3 \log_{x} (y)$

Step 1

Simplify each term.

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Step 1.1

Simplify $0.5 \log_{x} (y)$ by moving $0.5$ inside the logarithm.

$\log_{x} (y^{0.5}) + 5 \log_{x} (z) - 3 \log_{x} (y)$

Step 1.2

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

$\log_{x} (y^{0.5}) + \log_{x} (z^{5}) - 3 \log_{x} (y)$

Step 1.3

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

$\log_{x} (y^{0.5}) + \log_{x} (z^{5}) - \log_{x} (y^{3})$

Step 2

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

$\log_{x} (y^{0.5} z^{5}) - \log_{x} (y^{3})$

Step 3

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

$\log_{x} (\frac{y^{0.5} z^{5}}{y^{3}})$

Step 4

Move $y^{0.5}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\log_{x} (\frac{z^{5}}{y^{3} y^{- 0.5}})$

Step 5

Multiply $y^{3}$ by $y^{- 0.5}$ 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.

$\log_{x} (\frac{z^{5}}{y^{3 - 0.5}})$

Step 5.2

Subtract $0.5$ from $3$ .

$\log_{x} (\frac{z^{5}}{y^{2.5}})$