To evaluate $\log_{14}{17}$ using common logarithms ($\log$) or natural logarithms ($\ln$), we can apply the change of base formula:

$\log_b{a} = \frac{\log{a}}{\log{b}} \quad \text{or} \quad \log_b{a} = \frac{\ln{a}}{\ln{b}}$

Step 1: Apply the formula

$\log_{14}{17} = \frac{\log{17}}{\log{14}}$

Step 2: Calculate using a calculator

• $\log{17} \approx 1.2304$
• $\log{14} \approx 1.1461$

$\log_{14}{17} = \frac{1.2304}{1.1461} \approx 1.0737$

Final Answer:

$\boxed{1.0737}$

Would you like a detailed explanation of the change of base formula?

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Here are 5 related questions to expand on this topic:

1. How do you convert between natural logarithms and common logarithms?
2. Why is the change of base formula valid for all logarithms?
3. How can you use logarithms to solve exponential equations?
4. What is the difference between natural and common logarithms in applications?
5. How do logarithmic properties simplify complex expressions?

Tip: Always carry extra decimal places in intermediate steps to reduce rounding errors, then round only the final answer.