Precalculus Examples

$10 \log (\frac{a}{10^{- 16}}) - 10 \log (\frac{V}{10^{- 16}})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\frac{a}{10^{- 16}}$ as $a {(10^{- 16})}^{- 1}$ .

$10 \log (a {(10^{- 16})}^{- 1}) - 10 \log (\frac{V}{10^{- 16}})$

Step 1.2

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

$10 \log (a \cdot 10^{- 16 \cdot - 1}) - 10 \log (\frac{V}{10^{- 16}})$

Step 1.3

Rewrite $\log (a \cdot 10^{- 16 \cdot - 1})$ as $\log (a) + \log (10^{- 16 \cdot - 1})$ .

$10 (\log (a) + \log (10^{- 16 \cdot - 1})) - 10 \log (\frac{V}{10^{- 16}})$

Step 1.4

Use logarithm rules to move $- 16 \cdot - 1$ out of the exponent.

$10 (\log (a) - 16 \cdot - 1 \log (10)) - 10 \log (\frac{V}{10^{- 16}})$

Step 1.5

Multiply $- 16$ by $- 1$ .

$10 (\log (a) + 16 \log (10)) - 10 \log (\frac{V}{10^{- 16}})$

Step 1.6

Logarithm base $10$ of $10$ is $1$ .

$10 (\log (a) + 16 \cdot 1) - 10 \log (\frac{V}{10^{- 16}})$

Step 1.7

Multiply $16$ by $1$ .

$10 (\log (a) + 16) - 10 \log (\frac{V}{10^{- 16}})$

Step 1.8

Apply the distributive property.

$10 \log (a) + 10 \cdot 16 - 10 \log (\frac{V}{10^{- 16}})$

Step 1.9

Simplify $10 \log (a)$ by moving $10$ inside the logarithm.

$\log (a^{10}) + 10 \cdot 16 - 10 \log (\frac{V}{10^{- 16}})$

Step 1.10

Multiply $10$ by $16$ .

$\log (a^{10}) + 160 - 10 \log (\frac{V}{10^{- 16}})$

Step 1.11

Rewrite $\frac{V}{10^{- 16}}$ as $V {(10^{- 16})}^{- 1}$ .

$\log (a^{10}) + 160 - 10 \log (V {(10^{- 16})}^{- 1})$

Step 1.12

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

$\log (a^{10}) + 160 - 10 \log (V \cdot 10^{- 16 \cdot - 1})$

Step 1.13

Rewrite $\log (V \cdot 10^{- 16 \cdot - 1})$ as $\log (V) + \log (10^{- 16 \cdot - 1})$ .

$\log (a^{10}) + 160 - 10 (\log (V) + \log (10^{- 16 \cdot - 1}))$

Step 1.14

Use logarithm rules to move $- 16 \cdot - 1$ out of the exponent.

$\log (a^{10}) + 160 - 10 (\log (V) - 16 \cdot - 1 \log (10))$

Step 1.15

Multiply $- 16$ by $- 1$ .

$\log (a^{10}) + 160 - 10 (\log (V) + 16 \log (10))$

Step 1.16

Logarithm base $10$ of $10$ is $1$ .

$\log (a^{10}) + 160 - 10 (\log (V) + 16 \cdot 1)$

Step 1.17

Multiply $16$ by $1$ .

$\log (a^{10}) + 160 - 10 (\log (V) + 16)$

Step 1.18

Apply the distributive property.

$\log (a^{10}) + 160 - 10 \log (V) - 10 \cdot 16$

Step 1.19

Simplify $- 10 \log (V)$ by moving $10$ inside the logarithm.

$\log (a^{10}) + 160 - \log (V^{10}) - 10 \cdot 16$

Step 1.20

Multiply $- 10$ by $16$ .

$\log (a^{10}) + 160 - \log (V^{10}) - 160$

Step 2

Combine the opposite terms in $\log (a^{10}) + 160 - \log (V^{10}) - 160$ .

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Step 2.1

Subtract $160$ from $160$ .

$\log (a^{10}) + 0 - \log (V^{10})$

Step 2.2

Add $\log (a^{10})$ and $0$ .

$\log (a^{10}) - \log (V^{10})$

Step 3

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

$\log (\frac{a^{10}}{V^{10}})$