Problem 37 - Algebra for College Students

Question

Approximate each logarithm to three decimal places. $$ \log _{4} 1.6 $$

Step-by-Step Solution

Answer

\(\log_4 1.6 \approx 0.339\)

Step 1: Change of Base Formula

To approximate this logarithm, we will use the change of base formula. The change of base formula is given by:\[\log_b a = \frac{\log_c a}{\log_c b}\]where "c" is a base that makes the computation easier, often 10 or e (natural log). For this problem, we use base 10. So, we have:\[\log_4 1.6 = \frac{\log_{10} 1.6}{\log_{10} 4}\].

Step 2: Use a Calculator

Now we need to compute \(\log_{10} 1.6\) and \(\log_{10} 4\) using a calculator:1. \(\log_{10} 1.6 \approx 0.204\)2. \(\log_{10} 4 \approx 0.602\)

Step 3: Calculate the Approximate Value

We now divide the two results from Step 2 to find \(\log_4 1.6\):\[\log_4 1.6 = \frac{0.204}{0.602} \approx 0.339\].

Step 4: Final Approximation

According to the calculations, the value of \(\log_4 1.6\) is approximately 0.339 when rounded to three decimal places.

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool in mathematics. It is particularly useful when you're dealing with logarithms of bases that aren't easily handled with a calculator. Essentially, it allows us to convert a logarithm from one base to a more convenient common base, like 10 or e (natural logarithm). This is very advantageous because calculators usually have keys only for base 10 or natural logarithms.

Here's how it works:

• If you have a logarithm \(\log_b a\), you can use the change of base formula: \[\log_b a = \frac{\log_c a}{\log_c b}\]
• "c" is typically chosen to be 10 or e. These bases are more manageable with standard calculator functions.

By applying this formula, we can transform something like \(\log_4 1.6\) into an equation involving base 10 logarithms, as in: \[\log_4 1.6 = \frac{\log_{10} 1.6}{\log_{10} 4}\]. This makes the task of finding the logarithmic value more straightforward.

Logarithmic Approximation

Logarithmic approximation involves estimating the value of a logarithm to a desired degree of accuracy, often using a calculator. This is particularly useful when the exact values are either difficult or impossible to determine due to irrational numbers.

For example, when you approximate \(\log_{10} 1.6\) and \(\log_{10} 4\), you're finding values that are close enough for practical purposes, if not exact. The approximations, \(\log_{10} 1.6 \approx 0.204\) and \(\log_{10} 4 \approx 0.602\), help simplify otherwise complicated calculations and allow further mathematical operations like division or multiplication to be more manageable.

The idea is often to round these approximations to a specified number of decimal places, such as three decimal places in our example. This level of precision is usually sufficient for tasks where absolute accuracy isn't critical, but readability and ease of calculation are preferred.

Base 10 Logarithm

Base 10 logarithm, also known as the common logarithm, is frequently used in various scientific fields. Represented as \(\log_{10}\), it's a scale that is easy for humans to understand and compute. This is one reason calculators and many mathematical software tools offer direct functions for base 10 logarithms.

When calculating \(\log_{10} a\), you essentially find out what power you need to raise 10 to, to yield "a". For example, if \(\log_{10} 100 = 2\), this implies \(10^2 = 100\). Such calculations are intuitive, making base 10 logs ideal for approximations in diverse applications like engineering or physics.

• Use in Measure: Base 10 logarithms are used in measuring phenomena across scales, like the Richter scale for earthquakes.
• Calculator Readiness: Given that most calculators natively support base 10 calculations (the "log" function), it's often the go-to choice for quick computations.

Understanding and using \(\log_{10}\) effectively allows for efficient computation and problem-solving when dealing with exponential growth or decay.