Math Problem Statement

Use common logarithms or natural logarithms and a calculator to evaluate the expression. log Subscript 14 Baseline 17 Question content area bottom Part 1 Evaluate the expression. log Subscript 14 Baseline 17almost equals    enter your response here ​(Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as​ needed.)

Solution

To evaluate log1417\log_{14}{17} using common logarithms (log\log) or natural logarithms (ln\ln), we can apply the change of base formula:

logba=logalogborlogba=lnalnb\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

log1417=log17log14\log_{14}{17} = \frac{\log{17}}{\log{14}}

Step 2: Calculate using a calculator

  • log171.2304\log{17} \approx 1.2304
  • log141.1461\log{14} \approx 1.1461

log1417=1.23041.14611.0737\log_{14}{17} = \frac{1.2304}{1.1461} \approx 1.0737

Final Answer:

1.0737\boxed{1.0737}

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


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.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Change of Base Formula

Formulas

\log_b{a} = \frac{\log{a}}{\log{b}}
\log_b{a} = \frac{\ln{a}}{\ln{b}}

Theorems

Change of Base Formula

Suitable Grade Level

Grades 9-12