Problem 58 - Algebra and Trigonometry

Question

\(55-62\) . Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$ \log _{6} 92 $$

Step-by-Step Solution

Answer

\(\log_{6} 92 \approx 2.523717\).

Step 1: Apply the Change of Base Formula

Using the change of base formula, we can convert the logarithm with base 6 to a common logarithm (base 10) or natural logarithm (base e). The formula is: \[\log_{b} a = \frac{\log_{c} a}{\log_{c} b}\]Here, choose \(c\) as 10 (common log) or \(e\) (natural log). So, for \(\log_{6} 92\), it becomes:\[\log_{6} 92 = \frac{\log_{10} 92}{\log_{10} 6}\]or\[\log_{6} 92 = \frac{\ln 92}{\ln 6}\]

Step 2: Calculate Logarithms with Calculator

Use a calculator to find the logarithms. For common logarithms, calculate:\(\log_{10} 92\) and \(\log_{10} 6\). For natural logarithms, calculate \(\ln 92\) and \(\ln 6\). For common logs: \(\log_{10} 92 \approx 1.96379\)\(\log_{10} 6 \approx 0.77815\)For natural logs:\(\ln 92 \approx 4.52179\)\(\ln 6 \approx 1.79176\)

Step 3: Compute the Division

Using the calculated logarithms, divide as per the change of base formula. For common logs:\[\frac{\log_{10} 92}{\log_{10} 6} = \frac{1.96379}{0.77815} \approx 2.52372\]Using natural logs:\[\frac{\ln 92}{\ln 6} = \frac{4.52179}{1.79176} \approx 2.52372\]Both approaches should give the same result.

Step 4: Round the Result

Round the result from Step 3 to six decimal places, which is \(2.523717\). This is the value of \(\log_{6} 92\).

Key Concepts

Common Logarithm

The common logarithm is essentially a type of logarithm that uses base 10. It's quite common in mathematical and scientific calculations, hence its name. Common logarithms are often expressed as \( \log_{10} \) or simply \( \log \) when the base is understood to be 10. This makes them particularly useful when working on problems involving the powers of 10, which are prevalent in real-world situations.

One of the key advantages of the common logarithm is the ease of computation, as most scientific calculators come equipped with a dedicated "log" function to quickly perform these operations. It's also helpful in simplifying and solving exponential growth and decay problems.

For example, in the given problem, to find \( \log_{6} 92 \) using the change of base formula, you can convert it into a common logarithm:

• First, calculate \( \log_{10} 92 \approx 1.96379 \)
• Then, calculate \( \log_{10} 6 \approx 0.77815 \)

By dividing these values using the change of base formula, you simplify the calculation to find the result.

Natural Logarithm

Natural logarithms are another type of logarithm that are based on the mathematical constant \( e \), which is approximately equal to 2.71828. These logarithms are commonly used in more advanced mathematics and science, particularly where growth processes are modeled, such as in biology or finance.

The notation for natural logarithms is \( \ln \). They are highly significant in calculus and other mathematical fields due to their natural properties that make solving differential equations easier.

With the change of base formula, we can also use natural logarithms to find \( \log_{6} 92 \):

• Calculate \( \ln 92 \approx 4.52179 \)
• Calculate \( \ln 6 \approx 1.79176 \)

Now, by dividing these natural logs, we can also achieve the desired result. This showcases the flexibility of using the change of base formula with either common or natural logarithms, while ensuring you get the same final answer.

Logarithmic Functions

Logarithmic functions are a special class of mathematical functions that help in solving problems that deal with exponential relationships. They are essentially the inverses of exponential functions, meaning they allow you to "reverse" the process of exponentiation.

A logarithmic function can be written in the form \( y = \log_{b} x \), which tells you the power to which the base \( b \) must be raised to produce the number \( x \). This concept is central to the solution of the exercise, as it allows us to convert bases and solve logarithms using standard calculator functions like \( \log \) and \( \ln \).

The change of base formula \[ \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \] is pivotal because it enables us to evaluate logarithms of any base using either common or natural logarithms. This shows the power and adaptability of logarithmic functions in simplifying complex calculations and highlighting the relationship between exponential growth and logarithmic scale.

Logarithmic functions have a remarkable range of applications, from calculating the logarithmic decay of substances to solving equations involving orders of magnitude, making them indispensable in both academic and practical contexts.