If 3a=4,4b=5,5c=6,6d=7,7e=8 and 8f=9,thenthevalueoftheproduct\textit{abcdef }$ is
TITA Answer:
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Official Correct Answer: 2. ### Core Concept
The problem involves solving for exponents and then multiplying them. The key concept is the change of base formula for logarithms:
$$\log_{x} y = \frac{\ln y}{\ln x}$$
Additionally, the property of logarithms that allows the product of consecutive logarithms to telescope is used here.
### Step-by-Step Solution
1. **Express each variable using logarithms:**
\[
a = \log_{3} 4, \quad b = \log_{4} 5, \quad c = \log_{5} 6, \quad d = \log_{6} 7, \quad e = \log_{7} 8, \quad f = \log_{8} 9
\]
2. **Write the product \(abcdef\) using logarithms:**
\[
(\log_{3} 4) \times (\log_{4} 5) \times (\log_{5} 6) \times (\log_{6} 7) \times (\log_{7} 8) \times (\log_{8} 9)
\]
3. **Apply the change of base formula to each logarithm:**
\[
\left(\frac{\ln 4}{\ln 3}\right) \times \left(\frac{\ln 5}{\ln 4}\right) \times \left(\frac{\ln 6}{\ln 5}\right) \times \left(\frac{\ln 7}{\ln 6}\right) \times \left(\frac{\ln 8}{\ln 7}\right) \times \left(\frac{\ln 9}{\ln 8}\right)
\]
4. **Simplify by canceling intermediate terms:**
\[
\frac{\ln 9}{\ln 3}
\]
5. **Recognize the remaining logarithm:**
\[
\frac{\ln 9}{\ln 3} = \log_{3} 9
\]
6. **Simplify \(\log_{3} 9\):**
\[
\log_{3} 9 = \log_{3} 3^2 = 2
\]
### 30-Second Shortcut
Notice that each logarithm in the product cancels out the previous denominator, leaving only the first base and the last result. Thus, \(abcdef = \log_{3} 9 = 2\).
### Final Answer
Correct Answer: 2