Question

in exercises 23–30, condense the logarithmic expression. (see example 3.)

1. \\(\log _{6} 7 - \log _{6} 10\\)
2. \\(\ln 12 - \ln 4\\)
3. \\(6 \ln x + 4 \ln y\\)
4. \\(2 \log x + \log 11\\)
5. \\(\log _{5} 4 + \frac{1}{3} \log _{5} x\\)

Explanation:

Step1: Use log subtraction rule

$\log_b m - \log_b n = \log_b \frac{m}{n}$
$\log 7 - \log_{10} 10 = \log_{10} \frac{7}{10}$

Step2: Simplify $\log_{10}10$

$\log_{10}10 = 1$, so $\log 7 - 1$ (or $\log_{10}\frac{7}{10}$)

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Step1: Use log subtraction rule

$\ln m - \ln n = \ln \frac{m}{n}$
$\ln 12 - \ln 4 = \ln \frac{12}{4}$

Step2: Simplify the fraction

$\frac{12}{4}=3$, so $\ln 3$

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Step1: Use log addition rule

$\log_b m + \log_b n = \log_b (m \cdot n)$
$\ln 6 + \ln 4 = \ln (6 \times 4)$

Step2: Calculate product inside log

$6 \times 4=24$, so $\ln 24$

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Step1: Use log addition rule

$\log_b m + \log_b n = \log_b (m \cdot n)$
$\log 2 + \log 11 = \log (2 \times 11)$

Step2: Calculate product inside log

$2 \times 11=22$, so $\log 22$

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Step1: Rewrite coefficient as log exponent

$k\log_b m = \log_b m^k$
$\frac{1}{3}\log_5 x = \log_5 x^{\frac{1}{3}} = \log_5 \sqrt[3]{x}$

Step2: Use log addition rule

$\log_b m + \log_b n = \log_b (m \cdot n)$
$\log_5 4 + \log_5 \sqrt[3]{x} = \log_5 (4\sqrt[3]{x})$

Answer:

1. $\log_{10}\frac{7}{10}$ (or $\log 7 - 1$)
2. $\ln 3$
3. $\ln 24$
4. $\log 22$
5. $\log_5 (4\sqrt[3]{x})$