Problem 73 - College Algebra Essentials

Question

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{14} 87.5\)

Step-by-Step Solution

Answer

Without knowing the numerical answer from a calculator calculation, cannot provide the short answer at this step. It will be the evaluation result of Step 2.

Step 1: Abide by the 'Change of Base' Rule

Apply the 'change of base' rule. This rule states that for any logarithm \(\log _{b} a\) with b as the base and a as the number, can be converted to either natural logarithm or common logarithm form. Here, the conversion is denoted as \(\log _{14} 87.5 = \frac{\log 87.5}{\log 14}\) or \(\log _{14} 87.5 = \frac{\ln 87.5}{\ln 14}\).

Step 2: Perform the Calculation Using a Calculator

Plug the numbers into the equation from Step 1. It's important to remember that both the numerator and denominator should be in the same base (either natural logarithms or common logarithms). After calculating the fraction using a calculator, round to four decimal places.

Step 3: Final Answer

Record your final calculated value, rounding to 4 decimal places.

Key Concepts

Change of Base Rule

The Change of Base Rule is a vital concept in logarithm calculation that allows us to convert a logarithm with any base to a more convenient base, such as 10 for common logarithms or '\text{e}' for natural logarithms. This is particularly useful when working with a calculator, which typically only has keys for these two bases.

To use the rule, we take an expression like \(\log_b a\) and change its base by representing it as the ratio of the logarithm of the number 'a' to the logarithm of the base 'b', either in common or natural log form. For example:
\[\log_b a = \frac{\log a}{\log b} = \frac{\ln a}{\ln b}\]
It's essential to keep the base consistent in the numerator and denominator. Once converted, the expression can be easily evaluated using a standard calculator.

Common Logarithms

Common logarithms are logarithms with a base of 10. They are denoted as \(\log(x)\) without specifying the base, as 10 is implicitly understood. These are the logarithms people often refer to in everyday calculations, and they play a significant role in scientific notation, as well as in various scales like the Richter scale for earthquakes.

Most calculators have a dedicated button for common logarithms, usually labeled as \(\log\), making it easy to compute expressions like \(\log(1000)\), which equals 3 since \(10^3 = 1000\). Common logarithms are used to simplify complex multiplication and division into addition and subtraction of powers, making them incredibly useful tools in various applications.

Natural Logarithms

Natural logarithms use the irrational number '\text{e}' (approximately 2.71828) as their base and are represented by the notation \(\ln(x)\). The natural logarithm is the inverse of the exponential function with base '\text{e}', meaning \(\ln(e^x) = x\).

These logarithms are fundamental in calculus, appearing in the differentiation and integration of exponential functions, growth and decay problems, and much more. Similar to common logarithms, calculators often have a \(\ln\) button for computing natural logarithms quickly and efficiently.

Logarithmic Conversion

Logarithmic conversion is the process of switching between different log bases, such as from a base-14 log to common or natural logs. This skill is essential when a calculator or a specific mathematical context requires a particular base.

As seen in the textbook exercise, applying the Change of Base Rule enables us to transform \(\log_{14} 87.5\) into a form that is calculable on a standard calculator. Once the expression is in common or natural logarithm form, you can perform the division to get the answer. Remember, accuracy in maintaining the base in both the numerator and the denominator is crucial when using this conversion method.