Problem 37 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{1 / 5} 31 $$

Step-by-Step Solution

Answer

The value of the \(\log_{1/5} 31\) rounded to three decimal places after performing all the steps.

Step 1: Understanding Logarithm

The logarithm of a number is the exponent to which another fixed value (the base) must be raised to produce that number. Here, we need to calculate \(\log_{1/5} 31\) which means we need to identify the exponent to which \(1/5\) must be raised in order to get \(31\).

Step 2: Base Conversion

Natural logarithm or logarithm base e (\(\log_e\)), also known as ln, is more commonly used in calculators. Given a logarithm base b of x: \(\log_b x\), it can also be expressed as the ratio of the natural logarithm of x to the natural logarithm of b, which is \(\log_b x = \frac{\log_e x}{\log_e b}\). We can use this property to convert \(\log_{1/5} 31\) into an equivalent form that uses the natural logarithm.

Step 3: Substitute the Values

Now substitute \(b = 1/5\) and \(x = 31\) into \(\frac{\log_e x}{\log_e b}\), we have \(\log_{1/5} 31 = \frac{\log_e 31}{\log_e (1/5)}\)

Step 4: Use a Scientific Calculator

Use a scientific calculator to calculate \(log_e 31\) and \(log_e (1/5)\), then divide the results.

Step 5: Round the Result

You will get a number for the solution of the logarithm, round this to three decimal places.

Key Concepts

Base Conversion

Base conversion in logarithms involves changing the base of a logarithmic expression to another base, most commonly the natural logarithm or \e\. This is useful because scientific calculators typically provide direct functions for calculating natural logarithms. The relationship for base conversion is given by the formula:

• \[ \log_b a = \frac{\log_e a}{\log_e b} \]

Here, \( \log_e \) stands for the natural logarithm. So, when you have something like \( \log_{\frac{1}{5}} 31 \), you can convert it by using:

• \( \log_{\frac{1}{5}} 31 = \frac{\log_e 31}{\log_e (\frac{1}{5})} \)

This transformation allows one to compute logarithm expressions that might not readily fit the commonly available functions on a calculator.

Natural Logarithm

A natural logarithm is a logarithm to the base \e\, where \e\ is approximately 2.71828. The symbol for the natural logarithm is \(\ln\). It's a key concept in mathematics because the base \e\ arises naturally in many areas of mathematics, such as calculus and complex numbers.
When dealing with natural logarithms, the following properties are useful:

• \( \ln(1) = 0 \) because \e^{0} = 1\.
• \( \ln(e) = 1 \) by definition, since \e^1 = e\.
• \( \ln(xy) = \ln(x) + \ln(y) \) which is the logarithmic identity.

Understanding these properties can make it easier to manipulate and solve equations involving \ln\. In converting \(\log_{b} x\) to \(\ln\) form as shown earlier, we use the ratio of natural logarithms to change the base of the logarithm.

Scientific Calculator Usage

Using a scientific calculator efficiently can help in solving logarithmic expressions easily. Almost all scientific calculators have built-in functions for calculating logarithms including the common logarithm (\(\log_{10}\)) and the natural logarithm (\(\ln\)). Here’s a brief guide on how to use a scientific calculator for our exercise:

• First, use the \(\ln\) button on your calculator to find \(\ln 31\).
• Next, find \(\ln (1/5)\), again using the \(\ln\) button.
• After that, divide the value of \(\ln 31\) by \(\ln (1/5)\) to convert \(\log_{\frac{1}{5}} 31\) into a \ln\ form as per base conversion.
• Finally, round your result to three decimal places as needed.

This process makes it possible to solve complex logarithms using simple calculator operations, helping streamline finding the correct answer quickly.