Precalculus Examples

$2 (\log_{5} (a) + (\frac{1}{2}) \log_{5} (d)) - 3 \log_{5} (b) - 4 \log_{5} (c)$

Step 1

Simplify each term.

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Step 1.1

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

$2 (\log_{5} (a) + \log_{5} (d^{\frac{1}{2}})) - 3 \log_{5} (b) - 4 \log_{5} (c)$

Step 1.2

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

$2 \log_{5} (a d^{\frac{1}{2}}) - 3 \log_{5} (b) - 4 \log_{5} (c)$

Step 1.3

Simplify $2 \log_{5} (a d^{\frac{1}{2}})$ by moving $2$ inside the logarithm.

$\log_{5} ({(a d^{\frac{1}{2}})}^{2}) - 3 \log_{5} (b) - 4 \log_{5} (c)$

Step 1.4

Apply the product rule to $a d^{\frac{1}{2}}$ .

$\log_{5} (a^{2} {(d^{\frac{1}{2}})}^{2}) - 3 \log_{5} (b) - 4 \log_{5} (c)$

Step 1.5

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

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Step 1.5.1

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

$\log_{5} (a^{2} d^{\frac{1}{2} \cdot 2}) - 3 \log_{5} (b) - 4 \log_{5} (c)$

Step 1.5.2

Cancel the common factor of $2$ .

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Step 1.5.2.1

Cancel the common factor.

$\log_{5} (a^{2} d^{\frac{1}{2} \cdot 2}) - 3 \log_{5} (b) - 4 \log_{5} (c)$

Step 1.5.2.2

Rewrite the expression.

$\log_{5} (a^{2} d^{1}) - 3 \log_{5} (b) - 4 \log_{5} (c)$

Step 1.6

Simplify.

$\log_{5} (a^{2} d) - 3 \log_{5} (b) - 4 \log_{5} (c)$

Step 1.7

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

$\log_{5} (a^{2} d) - \log_{5} (b^{3}) - 4 \log_{5} (c)$

Step 1.8

Simplify $- 4 \log_{5} (c)$ by moving $4$ inside the logarithm.

$\log_{5} (a^{2} d) - \log_{5} (b^{3}) - \log_{5} (c^{4})$

Step 2

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

$\log_{5} (\frac{a^{2} d}{b^{3}}) - \log_{5} (c^{4})$

Step 3

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

$\log_{5} (\frac{\frac{a^{2} d}{b^{3}}}{c^{4}})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{5} (\frac{a^{2} d}{b^{3}} \cdot \frac{1}{c^{4}})$

Step 5

Combine.

$\log_{5} (\frac{a^{2} d \cdot 1}{b^{3} c^{4}})$

Step 6

Multiply $a^{2}$ by $1$ .

$\log_{5} (\frac{a^{2} d}{b^{3} c^{4}})$