Find the exact value of each expression.(a)log4(20) − log4(45) + log4(144)(b)log5(120) − log5(50) − log5(60)

(a) Break Down the Problem

1. We need to simplify the expression $\log_4(20) - \log_4(45) + \log_4(144)$.

(b) Relevant Concepts

1. We can use logarithmic properties:
   • $\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)$
   • $\log_b(m) + \log_b(n) = \log_b(m \cdot n)$

(c) Analysis and Detail

1. Start by combining the first two logarithms: $\log_4(20) - \log_4(45) = \log_4\left(\frac{20}{45}\right) = \log_4\left(\frac{4}{9}\right)$

2. Now include the third logarithm: $\log_4\left(\frac{4}{9}\right) + \log_4(144) = \log_4\left(\frac{4}{9} \cdot 144\right)$

3. Calculate $\frac{4}{9} \cdot 144$: $\frac{4 \cdot 144}{9} = \frac{576}{9} = 64$

4. Therefore, we have: $\log_4(64)$

5. Since $64 = 4^3$: $\log_4(64) = 3$

(d) Verify and Summarize

1. We have confirmed the calculations and simplified using logarithmic properties to find the exact value.

Final Answer

The exact value of $\log_4(20) - \log_4(45) + \log_4(144)$ is $3$.

(b) Break Down the Problem

1. We need to simplify the expression $\log_5(120) - \log_5(50) - \log_5(60)$.

(c) Relevant Concepts

1. Using the logarithmic properties mentioned above.

(d) Analysis and Detail

1. Combine the first two logarithms: $\log_5(120) - \log_5(50) = \log_5\left(\frac{120}{50}\right) = \log_5\left(\frac{12}{5}\right)$

2. Now include the third logarithm: $\log_5\left(\frac{12}{5}\right) - \log_5(60) = \log_5\left(\frac{12/5}{60}\right) = \log_5\left(\frac{12}{300}\right) = \log_5\left(\frac{1}{25}\right)$

3. Since $\frac{1}{25} = 5^{-2}$: $\log_5\left(\frac{1}{25}\right) = -2$

(e) Verify and Summarize

1. The calculations have been verified; thus, the steps and properties utilized are correct.

Final Answer

The exact value of $\log_5(120) - \log_5(50) - \log_5(60)$ is $-2$.