Problem 90 - College Algebra

Question

Explain how to use your calculator to find \(\log _{14} 283\).

Step-by-Step Solution

Answer

The value of \(\log _{14} 283\) is the result of dividing the value of \(\log _{10} 283\) by the value of \(\log _{10} 14\) in the calculator

Step 1: Understanding the Problem

The problem asks to find \(\log _{14} 283\). The base of the logarithm is 14, and the number inside the logarithm is 283. The calculator to be used may not have a function to compute the logarithm with base 14. Therefore, the change of base formula will be utilized.

Step 2: Utilize the Change of Base Formula

The change of base formula allows the conversion of \(\log _{14} 283\) to a base that the calculator can handle. A common base used is base 10. Using the change of base formula, the problem can be rewritten as \( \frac {\log _{10} 283}{ \log _{10} 14}\)

Step 3: Solve using the Calculator

Use the calculator to evaluate both \(\log _{10} 283\) and \(\log _{10} 14\), and then divide the two results. The result is the value of \(\log _{14} 283\)

Key Concepts

Understanding Logarithms

A logarithm is essentially an exponent that tells us how many times we need to multiply a base number to get a certain value. It is written in the form of \( \text{log}_b (a) \), where \( b \) is the base, and \( a \) is the number we want to reach through repeated multiplication of the base. For instance, if we have \( \text{log}_2 (8) \), this would translate to asking the question, 'how many twos do we need to multiply together to get eight?' The answer is three, because \( 2 \times 2 \times 2 = 8 \).

Understanding the logarithmic relationship can be instrumental not just in algebra, but in many fields, including science, engineering, and even finance. Logarithms help simplify multiplication and division to addition and subtraction, offering a shortcut to dealing with large numbers or small, bothersome decimals.

Calculator Usage in Algebra

Calculators are incredible tools that assist in performing algebraic operations efficiently and accurately, including logarithmic calculations. Most scientific calculators have a dedicated logarithm function key, often labeled as \( \text{LOG} \) for the common or base-10 logarithm, and \( \text{LN} \) for the natural or base-\( e \) logarithm. When dealing with bases that are not 10 or \( e \), we can use the change of base formula to transform the expression into a form that the calculator can understand.

The change of base formula is expressed as \( \text{log}_b(a) = \frac{\text{log}_c(a)}{\text{log}_c(b)} \), where \( c \) can be any number, but using 10 or \( e \) makes the calculation straightforward due to the calculator's built-in functions. By using this formula, students can extend the use of their calculators to solve logarithms of any base, making them a versatile tool in mathematics education and beyond.

Executing Logarithmic Calculations

When performing logarithmic calculations, such as finding \( \text{log}_{14}(283) \), we turn to the change of base formula to convert the base to one that our calculator can compute. The choice can be either base 10 or base \( e \), depending on which keys are available. After applying the change of base formula, you end up with a fraction of two logarithms, which you can then calculate separately. In this instance, you would first find \( \text{log}_{10}(283) \), then \( \text{log}_{10}(14) \), and finally divide the former by the latter.

The calculator simplifies this process, as most models will directly compute these common logarithms. Remember to double-check your inputs to avoid any unintended errors. Mastering the use of the change of base formula and your calculator together will enable you to tackle a whole host of logarithmic problems with ease and confidence.