Problem 8 - Precalculus: A Concise Course

Question

Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{3} 47$$

Step-by-Step Solution

Answer

So, the logarithm \(\log _{3} 47\) can be rewritten as \(\frac{\log_{10} 47}{\log_{10} 3}\) in terms of common logarithms and as \(\frac{\ln 47}{\ln 3}\) in terms of natural logarithms.

Step 1: Apply the change of base formula with base 10

We apply the change of base formula to rewrite \(\log _{3} 47\) in terms of common logarithms (base 10). We follow the formula \(\log_b a = \frac{\log_c a}{\log_c b}\) with \(a = 47\), \(b = 3\) and \(c = 10\). Therefore, \(\log _{3} 47\) becomes \(\frac{\log_{10} 47}{\log_{10} 3}\).

Step 2: Simplify the ratio of logarithms

We cannot simplify the expression further because logarithms of 47 and 3 are irrational numbers. Therefore, \(\frac{\log_{10} 47}{\log_{10} 3}\) is the simplified ratio of common logarithms.

Step 3: Apply the change of base formula with base e

Now, we rewrite \(\log _{3} 47\) in terms of natural logarithms (base \(e\)). We apply the change of base formula, this time letting \(c = e\). Thus, \(\log _{3} 47\) becomes \(\frac{\ln 47}{\ln 3}\).

Step 4: Simplify the ratio of natural logarithms

We cannot simplify the expression further because natural logarithms of 47 and 3 are irrational numbers. Therefore, \(\frac{\ln 47}{\ln 3}\) is the simplified form of natural logarithms.

Key Concepts

Change of Base Formula

Logarithms can often be tricky to work with, especially when dealing with different bases. Luckily, the change of base formula is a handy tool that helps to convert logarithms from one base to another. This can make calculations simpler and more manageable. The formula is written as follows:

• \( \log_b a = \frac{\log_c a}{\log_c b} \)

This means if you have a logarithm \( \log_b a \), you can rewrite it in terms of any other base \( c \) by dividing the logarithm of \( a \) to the new base by the logarithm of \( b \) to the same base. You can choose base 10 or the natural base \( e \), which lead us into our next topics: common and natural logarithms.For instance, if you need to calculate \( \log_3 47 \), converting it using the base 10 simplifies the task significantly, as shown below:

• \( \log_3 47 = \frac{\log_{10} 47}{\log_{10} 3} \)

This conversion is particularly useful when using calculators, as most only compute logarithms base 10 and base \( e \).

Common Logarithms

Common logarithms are logarithms that use base 10. They are widely used in scientific calculations. The notation is quite straightforward: \( \log_{10} a \) can simply be written as \( \log a \). Simply using \( \log \) to imply base 10 is a common mathematical shorthand.One reason for the pervasive use of common logarithms is their simplicity and the fact that they are easily programmable into calculators. In real-world applications, many systems and phenomena scale by powers of 10, making the common logarithm very useful.Let's look at an example:

• To rewrite \( \log_3 47 \) using common logarithms, apply the formula: \( \log_3 47 = \frac{\log_{10} 47}{\log_{10} 3} \).

Remember, common logarithms simplify working on problems that involve exponential growth or scientific measurements.

Natural Logarithms

Natural logarithms are based on the mathematical constant \( e \), which is approximately equal to 2.71828. The symbol for natural logarithms is \( \ln \). Natural logarithms are important in mathematics, especially in calculus and scientific contexts because they relate directly to exponential growth processes.The change of base formula can also be applied to convert logarithms to a natural base:

• For example, \( \log_3 47 \) becomes \( \frac{\ln 47}{\ln 3} \).

This conversion is very handy as it allows us to handle calculations involving exponentials rather efficiently.Natural logarithms are particularly significant when it comes to derivative and integral calculations, making them a crucial aspect of both theoretical and applied mathematics.Whether you are dealing with finance, physics, or another domain that involves growth processes, natural logarithms provide a powerful tool for analysis and problem-solving.