Problem 46 - 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. $$\log _{5} 0.65$$

Step-by-Step Solution

Answer

Using the change-of-base formula, \(\log_5 0.65\) can be computed and rounded off to four decimal places.

Step 1: Understand the Change-of-Base Formula

The change-of-base formula is written as \(\log_b a = \frac{\log a}{\log b}\), where \(\log a\) and \(\log b\) can be of any base that the calculator allows, usually 10 or \(e\).

Step 2: Apply the Change-of-Base Formula

Plugging in the given values from \(\log_5 0.65\) to the formula, it will become \(\frac{\log 0.65}{\log 5}\), where \(\log\) is base 10, which is compatible with most calculators.

Step 3: Compute using a Calculator and Round Off

Divide the logarithm of 0.65 by the logarithm of 5 using a calculator. The answer should be rounded to four decimal places.

Key Concepts

Understanding Logarithms

Logarithms are the mathematical operations that are the inverse of exponentiation. They tell us the power to which a base number, usually written as \(b\), must be raised to produce a given number. If we say \(\log_b a = c\), it means the base \(b\) raised to the power \(c\) equals \(a\).

• Example: If \(2^3 = 8\), then \(\log_2 8 = 3\).
• Logarithms simplify complex multiplications into manageable additions. They are widely used in many areas, such as in scientific calculations, to deal with very large or very small numbers.

When working with logarithms, understanding the essential properties like the product, quotient, and power rules is crucial. These properties help in simplifying complex expressions and calculating values more easily.

Exploring Base 10 Logarithms

The base 10 logarithm, also known as the common logarithm, is represented as \(\log\) without specifying the base explicitly (\(\log_{10}\) can be written as \(\log\)). They are widely used due to their convenience and are often assumed when no base is mentioned.

• This type of logarithm is used extensively in scientific applications and in any context where the decimal system is predominant.
• Because of their prevalence, most calculators have a dedicated button for base 10 logarithms.

Understanding how to convert between different bases using the change-of-base formula is key when working with non-decimal bases. This allows calculations using any base to be computed as if they were using base 10.

Calculator Usage for Logarithms

Using a calculator to compute logarithms simplifies the process, especially when dealing with the change-of-base formula. Here's how you can perform this on a standard scientific calculator:

• Enter the number for which you are calculating the logarithm (e.g., 0.65).
• Press the \(\log\) button to get the logarithm of the number to base 10.
• Repeat the above steps for the base of the logarithm (e.g., 5).
• Divide the result of the first logarithm by the result of the second to get the value of \(\log_5 0.65\).
• Round off the result to the required number of decimal places.

Rounding accurately, typically to four decimal places, ensures precision, especially important for scientific or engineering calculations. Remember to closely follow each step to ensure accurate results.