Chapter 10: Problem 31
In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{4} 7 $$
Short Answer
Expert verified
The value of \( \log_{4}7 \) to three decimal places is approximately 1.404 if we consider base e, and 1.403 if we consider base 10.
Step by step solution
01
Understanding the Problem
We are given the logarithm with base 4 for the number 7 and need to evaluate its value. We realize that most calculators use base 10 or base e (natural log) for their logarithm computations. However, we are dealing here with base 4. This requires us to use the 'Change of Base' formula which is \(log_b(a) = \frac{log_c(a)}{log_c(b)}\) where c can be any positive number different from 1.
02
Apply the Change of Base Formula
Applying the change of base formula to the given logarithm, we get \(\log_{4}7 = \frac{\log_{10} 7}{\log_{10} 4}\) or \(\log_{4}7 = \frac{ln 7}{ln 4}\). We can use any base but here we chose 10 and e for demonstration.
03
Calculate Using a Calculator
Now, we calculate the values using a calculator. We are asked to round our answer to three decimal places. Assuming base 10, the calculation would be \(\log_{4}7 = \frac{\log_{10} 7}{\log_{10} 4} \approx 1.403\). Assuming base e, the calculation would be \(\log_{4}7 = \frac{ln 7}{ln 4} \approx 1.404\).
04
The Final Answer
The value of \(\log_{4}7\) rounded to three decimal places is approximately 1.403 if we consider base 10, and 1.404 if we consider base e. Here, we can see that the result is not much affected by the change in the base. So, the choice of base can depend on the situation, but for most cases, we can use either base 10 or base e.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Change of Base Formula
The 'Change of Base' formula is a vital tool when working with logarithms, especially when the base of the logarithm isn't compatible with your calculator, which typically only has option for base 10 (common logarithm) and base e (natural logarithm). To address a problem like evaluating \(\log_{4}7\), we invoke the 'Change of Base' formula: \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\), where 'a' is the value you're taking the logarithm of, 'b' is the original base, and 'c' can be any base that your calculator can compute.
Let's break this formula down to understand why it's helpful. When a calculator doesn't have a direct function for \(\log_4\), use base 10 or e instead. So, \(\log_4\) can be expressed in terms of \(\log_{10}\) or \(\ln\) which stand for base 10 or e, respectively. This formula simply means that the logarithm of a number with any base can be calculated by dividing the logarithm of the number by the logarithm of the new base, both computed with the same base that your calculator does support.
Here are some key points:
Let's break this formula down to understand why it's helpful. When a calculator doesn't have a direct function for \(\log_4\), use base 10 or e instead. So, \(\log_4\) can be expressed in terms of \(\log_{10}\) or \(\ln\) which stand for base 10 or e, respectively. This formula simply means that the logarithm of a number with any base can be calculated by dividing the logarithm of the number by the logarithm of the new base, both computed with the same base that your calculator does support.
Here are some key points:
- The choice of 'c' depends on your calculator's capabilities.
- The 'Change of Base' formula shows that logarithms are proportional, allowing the use of any base to compute them.
- It ensures that you're never limited by your tools, as you can convert to a base that your calculator can handle.
Calculator Computation
Calculator computation for evaluating logarithms is incredibly straightforward once you've mastered the 'Change of Base' formula. If your calculator has buttons for \(\log\) (base 10) and \(\ln\) (base e), you're all set to calculate logarithms for any base. Using \(\log_{4}7\), you apply the formula and compute either \(\frac{\log_{10} 7}{\log_{10} 4}\) or \(\frac{ln 7}{ln 4}\) depending on the features of your calculator.
Considering the example at hand:
Modern calculators might have features that allow direct input of any base for the logarithm function. If not, using the 'Change of Base' formula coupled with either \(\log\) or \(\ln\) button provides the needed workaround.
Considering the example at hand:
- First, calculate \(\log_{10} 7\) or \(ln 7\).
- Second, calculate \(\log_{10} 4\) or \(ln 4\).
- Finally, divide the first result by the second to get your answer.
Modern calculators might have features that allow direct input of any base for the logarithm function. If not, using the 'Change of Base' formula coupled with either \(\log\) or \(\ln\) button provides the needed workaround.
Rounding Decimal Places
Rounding decimal places is an essential aspect of presenting your answer in a clear and concise manner, especially when dealing with irrational numbers or numbers that produce long decimal tails. While the example of \(\log_{4}7\) was calculated to be approximately 1.403 for base 10 and 1.404 for base e, these are rounded figures.
Here's what you should remember about rounding:
Here's what you should remember about rounding:
- Rounding to a certain number of decimal places means adjusting the number to the nearest value that contains that many digits to the right of the decimal point.
- The convention is to round up if the next digit is 5 or more, and to round down if it's less than 5.
- In our example, we are asked to round to three decimal places, which ensures a balance between accuracy and simplicity.