Problem 54 - Intermediate Algebra

Question

For Problems \(54-61\), perform the following calculations and express answers to the nearest hundredth. (These calculations are in preparation for our work in the next section.) $$ \frac{\log 7}{\log 3} $$

Step-by-Step Solution

Answer

The result is approximately 1.77.

Step 1: Understand the Expression

The expression \( \frac{\log 7}{\log 3} \) involves logarithms of numbers 7 and 3. The expression can be interpreted as the logarithm of 7 to the base 3. We will calculate it using common or natural logarithms, which default to base 10 or \(e\), respectively.

Step 2: Use a Calculator to Find Logarithms

Calculate the logarithm of 7 and the logarithm of 3 using a scientific calculator. For common logarithms (base 10), this would be \( \log_{10} 7 \approx 0.8451 \) and \( \log_{10} 3 \approx 0.4771 \).

Step 3: Compute the Quotient

Divide \( \log 7 \) by \( \log 3 \) using the computed values: \[ \frac{0.8451}{0.4771} \approx 1.77085 \].

Step 4: Round to the Nearest Hundredth

Round the result of the division to the nearest hundredth. The value \( 1.77085 \) rounded to the nearest hundredth is \( 1.77 \).

Key Concepts

Understanding Base 10 Logarithms

Base 10 logarithms, commonly denoted as \( \log_{10} \), are logarithms with a base of 10. In simpler terms, for a number \( x \), the base 10 logarithm answers the question, "To what power should we raise 10 to get \( x \)?" For instance, since \( 10^2 = 100 \), \( \log_{10}100 = 2 \). Base 10 logarithms are very useful in fields like science and engineering where scales can vary greatly.
To solve problems involving base 10 logarithms, you often need a calculator or a logarithm table because they're not typically simplified to neat integers. In practical applications, understanding the properties of logarithms, such as the product, quotient, and power rules, can simplify calculations and allow for more complex problem-solving. Knowing these can also help with expressing logarithms of one base in terms of another base, which often simplifies sharing results and communicating in standardized scientific contexts.

Utilizing a Scientific Calculator for Logarithms

A scientific calculator is equipped with functions that help in performing complex calculations like logarithms. When calculating logarithms such as \( \log_{10} 7 \), you'll typically find a button labeled "log" that defaults to base 10. When using a calculator:

• Ensure it is in the correct mode if multiple logarithmic bases are available.
• Enter the numerical value you need the logarithm for, then hit the "log" button.
• Record the resulting value, as these are crucial for further operations such as division or addition in logarithmic expressions.

Being familiar with your calculator's functions and how to operate them is crucial, especially when values need precision—like rounding to the nearest hundredth as in this exercise. Always double-check that numbers have been entered correctly and results seem reasonable, which helps avoid simple yet common errors.

Performing Arithmetic Operations with Logarithms

Arithmetic operations with logarithms, such as addition, subtraction, multiplication, and division, follow a unique set of rules due to the properties of logarithms. For the expression \( \frac{\log 7}{\log 3} \), division is the key operation. This is because \( \frac{\log a}{\log b} \) can also be interpreted as the logarithm of \( a \) to the base \( b \).

• First, calculate each logarithm individually using a scientific calculator.
• Next, perform the division as you would with any fraction: divide the first logarithm by the second.
• Ensure results are rounded appropriately, which is especially important when the problem specifies degrees of precision, such as to the nearest hundredth.

Understanding the logarithmic properties can simplify these operations. For instance, knowing that \( \log(a/b) = \log a - \log b \) can help with calculations involving subtraction. Applying these properties can make solving problems quicker and more intuitive, once you understand how to translate basic operations into logarithmic contexts.