Problem 43 - Precalculus : Building Concepts and Connections

Question

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{3} 1.25$$

Step-by-Step Solution

Answer

The value of \( \log_{3} 1.25 \) rounded to four decimal places is 0.2937.

Step 1: Write down the change-of-base formula

The change-of-base formula is a mathematical rule for changing the base of a logarithm. The formula is as follows: \[\log_{b} a = \frac{\ln a}{\ln b}\]where \(ln\) is the natural logarithm (base e). The numerator and the denominator can be any base, but using \(ln\) provides more accurate results and is already built in the calculator for easy computation.

Step 2: Apply the change-of-base formula

Substitute the values of a and b from the question (\(log_{3} 1.25\)) into the formula:\[\log_{3} 1.25 = \frac{\ln 1.25}{\ln 3}\]

Step 3: Calculate the values

Use a scientific calculator to find the values of \(\ln 1.25\) and \(\ln 3\), then divide the former by the latter.

Step 4: Round off the result

After evaluating the result, round it off to four decimal places as per the instructions provided in the question.

Key Concepts

Understanding Logarithms

A logarithm answers the question: 'To what exponent must we raise the base to get a certain number?' For example, in the expression \(\log_b a\), 'a' is the number you're trying to break down, and 'b' is the base. Remember, the base 'b' must always be a positive real number other than 1, and 'a' must always be a positive real number.

The concept of logarithms is crucial for understanding phenomena that grow exponentially or decay, as well as for solving equations where the unknown is an exponent, which is quite common in fields like physics, engineering, and finance.

To provide more clarity, let's illustrate with a simple example. If we have \(\log_2 8\), we are really asking 'What power do we raise 2 to get 8?' and the answer is 3, since \(2^3 = 8\). This demonstrates the inverse relationship between logarithms and exponentiation.

The Natural Logarithm: A Special Logarithm

The natural logarithm, denoted as \(\ln\), has a special base 'e', where 'e' is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm is the inverse of the exponential function \(e^x\), which means \(\ln(e^x) = x\) and \(e^{\ln x} = x\).

In more practical terms, the natural logarithm is used extensively in calculus, physics, and computer science due to its unique properties. For instance, in calculus, the rate of change of \(\ln x\) is \(1/x\), which makes computations involving growth and decay models more manageable.

Coming back to the exercise, when using the change-of-base formula \(\log_{b} a = \frac{\ln a}{\ln b}\), we are essentially re-expressing the logarithm in terms of natural logarithms because \(\ln\) functions are often pre-programmed into calculators, making it easier to evaluate logarithms with bases other than 10 or 'e' right at our fingertips.

Using a Scientific Calculator for Logarithms

Scientific calculators are equipped with functions to handle logarithms efficiently, especially the common log (base 10) and the natural log (base e). To solve logarithmic problems with different bases, you can utilize the change-of-base formula.

When using a scientific calculator, follow these steps: first, make sure the calculator is in the correct mode for your computation (for most logarithms, this will just be the usual computation mode). Then, input the values directly into the change-of-base formula as described in Step 2 of the exercise solution. Here's a quick guide to follow:

• Locate the \(\ln\) (natural log) button on your calculator.
• Enter the value of 'a', the number you're taking the logarithm of, and press \(\ln\).
• Do the same for 'b', the base of the logarithm, and then divide the two results.

After calculating, remember to round your answer as instructed. This method turns a potentially confusing logarithmic calculation into a more straightforward process, making scientific calculators invaluable tools in many mathematical and scientific applications.