Problem 33 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{2} 48 $$

Step-by-Step Solution

Answer

The evaluated value of \( \log_{2} 48 \) is approximately 5.585.

Step 1: Identifying the change of base formula

The change of base formula allows to compute the log with base 2 using the natural logarithm function. This formula is expressed as: \( \log_{b} a= \frac{\ln a}{\ln b} \).

Step 2: Substituting in the change of base formula

By substituting the values into the formula, we get: \( \log_{2} 48= \frac{\ln 48}{\ln 2} \).

Step 3: Evaluating the logarithm using a calculator

The natural logarithm of 48 and 2 can be respectively found using a calculator's 'ln' function, then divide the results to find the answer. Remember to round to 3 decimal places for your final answer.

Key Concepts

Change of Base Formula

Understanding logarithms often involves changing bases to make calculations easier. The Change of Base Formula is incredibly helpful in this regard. This formula allows you to convert any logarithm to a base that is easier to calculate. For instance, in the given exercise where we evaluated \( \log_{2} 48 \), we used the base change formula: \[ \log_{b} a = \frac{\ln a}{\ln b} \] This formula lets us switch from a base-2 logarithm to the natural logarithm \( \ln \) using base \( e \), which most calculators can easily compute.By substituting known values of \( \ln a \) and \( \ln b \), you simplify the calculation. The formula gives a means to transform the logarithm into a calculation of division between two natural logs. So essentially, no matter the base you start with, you can always use the Change of Base Formula to make your tasks simpler.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a log with base \( e \), where \( e \approx 2.718 \). It's one of the most used bases in logarithmic calculations. This base is special because it frequently occurs in calculus, solving continuous growth problems, and various exponential equations. Calculators have a specific key for the natural logarithm because it is used so often. To grasp how the natural log assists us, consider the exercise that involved calculating \( \ln 48 \) and \( \ln 2 \). By knowing how to use the calculator correctly for these values, and applying the change of base formula, you get the needed log value that's otherwise complex to evaluate with base 2 directly. Understanding the nature of \( \ln \) helps demystify why it's preferred in many mathematical transformations and practical applications.

Rounding Decimals

Rounding decimals is a crucial part of providing precise, legible results when dealing with irrational numbers and fractions in logarithmic calculations. To round a number to three decimal places, it means you must look at the fourth decimal to determine if the third decimal should round up or stay the same. For instance, when computing \( \log_{2} 48 \) after substitution into the change of base formula, the final result needs rounding after division. Rounding not only simplifies the appearance of long decimals but ensures consistency and comparability between results. Remember, the rules for rounding are:

• If the digit after your final destination digit (here, the third decimal) is 5 or more, round up.
• If it's less than 5, round down or leave it as is.

Rounding helps mathematicians communicate numbers unambiguously and maintains precision where needed in practical scenarios.