Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

Simplify $- 2 \log_{5} (2)$ by moving $2$ inside the logarithm.

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 1.4

Simplify $\frac{1}{2} \log_{5} (\frac{y}{4})$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 1.5

Apply the product rule to $\frac{y}{4}$ .

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

Step 1.6

Simplify the denominator.

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Step 1.6.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.6.2

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

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

Step 1.6.3

Cancel the common factor of $2$ .

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Step 1.6.3.1

Cancel the common factor.

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

Step 1.6.3.2

Rewrite the expression.

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

Step 1.6.4

Evaluate the exponent.

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

Step 2

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

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

Step 3

Combine ${(x - 3)}^{3}$ and $\frac{y^{\frac{1}{2}}}{2}$ .

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

Step 4

Reorder factors in $\log_{5} (\frac{{(x - 3)}^{3} y^{\frac{1}{2}}}{2})$ .

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