Problem 96 - Elementary and Intermediate Algebra

Question

Use the change-of-base formula to find logarithm to four decimal places. \(\log _{7} 3\)

Step-by-Step Solution

Answer

\(\log_7 3 \approx 0.5646\)

Step 1: Understand the Change of Base Formula

The change of base formula for logarithms is \(\log_b a = \frac{\log_c a}{\log_c b} \), where \(c\) is any positive number different from 1. It is often convenient to use base 10 or base \(e\) (natural logarithm).

Step 2: Apply the Change of Base Formula Using Base 10

To find \( \log_7 3 \), apply the formula: \( \log_7 3 = \frac{\log_{10} 3}{\log_{10} 7} \).

Step 3: Calculate \( \log_{10} 3 \) and \( \log_{10} 7 \)

Using a calculator, find \( \log_{10} 3 \approx 0.4771 \) and \( \log_{10} 7 \approx 0.8451 \).

Step 4: Divide the Logarithmic Values

Now, substitute the approximate values into the formula: \( \log_7 3 = \frac{0.4771}{0.8451} \).

Step 5: Simplify the Division

Perform the division: \( \frac{0.4771}{0.8451} \approx 0.5646 \). Hence, \( \log_7 3 \approx 0.5646 \).

Key Concepts

Logarithms

A logarithm is a mathematical function that helps answer the question: "To what power must a given base be raised, in order to produce a certain number?" Essentially, if you have an equation of the form \(b^n = a\), where \(b\) is the base and \(a\) is the desired number, the logarithm \(\log_b a = n\).
To put it simply, logarithms make it easier to work with very large or very small numbers by transforming multiplication into addition and division into subtraction. This is because each multiplication or division operation can be transformed into the sum or difference of logarithms.
Logarithms find applications in various fields such as science, engineering, and computer science, especially where exponential growth or decay is present. They are crucial for simplifying complex calculations and are the foundation of many important mathematical concepts.

Base 10 Logarithm

Base 10 logarithms, also known as common logarithms, are logarithms with 10 as the base. So, when we say \(\log_{10} a\), we are asking: "To what power must 10 be raised to result in \(a\)?"
Common logarithms are frequently used because they relate directly to our number system, which is decimal (base 10). This makes them particularly handy when discussing orders of magnitude in everyday contexts like population growth and sound intensity.

• For example, \(\log_{10} 1000 = 3\) because \(10^3 = 1000\).
• Calculation: Use a calculator as most also default to base 10 for any unexplained \(\log\).

Being able to switch from arbitrary bases to base 10 using the change-of-base formula enables us to compute logarithms that are otherwise complicated to calculate.

Natural Logarithm

Natural logarithms are logarithms with base \(e\), which is an important mathematical constant approximately equal to 2.71828. The natural logarithm of \(a\) is denoted as \(\ln a\), and it answers the question "To what power must \(e\) be raised to yield \(a\)?"
Unlike base 10 logarithms, natural logarithms are more commonly used in higher mathematics and natural sciences, particularly in calculus and the analysis of exponential growth and decay processes.

• Example: \(\ln e = 1\) because \(e^1 = e\).
• Basic property: \(\ln 1 = 0\) since \(e^0 = 1\).

They also play a significant role in solving complex equations and are essential for understanding compounding in contexts such as continuous compounding in finance or decay rates in radioactive substances.