Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms in Problems 33-36. \(\log _{5} 12\)

Step-by-Step Solution

Answer

\( \log_5 12 \approx 1.5441 \)

Step 1: Convert Logarithm to Natural Logarithms

The problem requires the use of the formula \( \log_{a} x = \frac{\ln x}{\ln a} \) to find the value of \( \log_5 12 \). Here, \( a = 5 \) and \( x = 12 \). Begin by expressing the equation as \[ \log_5 12 = \frac{\ln 12}{\ln 5} \].

Step 2: Calculate Natural Logarithms

Evaluate \( \ln 12 \) and \( \ln 5 \) using a calculator. \( \ln 12 \approx 2.4849 \) and \( \ln 5 \approx 1.60944 \).

Step 3: Divide the Natural Logarithms

Use the values obtained to compute \( \log_5 12 \): \[ \log_5 12 = \frac{2.4849}{1.60944} \approx 1.5441 \].

Step 4: Conclude the Calculation

The calculation \( \frac{\ln 12}{\ln 5} \) gives the value of \( \log_5 12 \) as approximately \( 1.5441 \). This is the final solution.

Key Concepts

Logarithms

Logarithms are a mathematical concept allowing you to solve equations involving exponential growth or decay. Essentially, a logarithm answers the question, "To what power must a particular base be raised to get a certain number?" For example, in the equation \( \, log_{10} 100 = 2 \, \), the base is 10, and it is raised to the power of 2 to get 100. The notation \( \, log_{a} x \, \) represents the power or exponent that the base \( \, a \, \) must be raised to in order to equal \( \, x \, \).

Logarithms can have different bases. The most common bases are 10 and \( \, e \, \), where \( \, e \, \) is the base of natural logarithms. With a variety of applications, logarithms are crucial in fields such as science and engineering.

Natural Logarithms

Natural logarithms, often represented as \( \, \ln x \, \), use the constant \( \, e \, \) (approximately equal to 2.71828) as their base. This special base is derived from limiting processes found in calculus and has unique properties that make natural logarithms integral to calculus.

In many natural growth and decay processes, \( \, e \, \) naturally arises due to its properties, especially in equations involving continuously compounding interest or understanding population dynamics. Consequently, knowing how to calculate and interpret natural logarithms is vital to analyzing such situations.

Change of Base Formula

The change of base formula is a useful tool when you need to calculate a logarithm with a base other than those commonly provided by calculators, which typically include only base 10 and the natural logarithm base \( \, e \, \).

This formula is expressed as \( \, \log_{a} x = \frac{\ln x}{\ln a} \, \).

This allows you to use natural logarithms, which are more accessible on calculators, to find logarithms of any base by simply converting between bases. This transformation makes calculations simpler and more consistent, especially when programming or using digital tools that do not support arbitrary bases.

Calculations with Natural Logs

Performing calculations using natural logarithms involves a couple of straightforward steps. Here is a streamlined guide:

• Identify the values of \( \, x \, \) and the base \( \, a \, \), which you might need to convert using the change of base formula.
• Compute \( \, \ln{x} \, \) and \( \, \ln{a} \, \) using a calculator, wherein these values represent the natural logarithms of \( \, x \, \) and \( \, a \, \) respectively.
• Divide the two results, \( \, \frac{\ln x}{\ln a} \, \), to arrive at the logarithm of your given base.

This process is exemplified in calculating \( \, \log_5 12 \, \), where natural logarithms are calculated as \( \, \ln 12 \approx 2.4849 \, \) and \( \, \ln 5 \approx 1.60944 \, \). Dividing these natural logs yields \( \, \log_5 12 \approx 1.5441\,. \)