Problem 61 - 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.) $$ \frac{\log 5}{3 \log 1.07} $$

Step-by-Step Solution

Answer

The answer is approximately 7.93.

Step 1: Understand the Expression

The expression given is \( \frac{ \log 5}{3 \log 1.07} \). It requires us to first compute the values of \( \log 5 \) and \( \log 1.07 \) using a calculator, and then perform the division and multiplication.

Step 2: Calculate \( \log 5 \)

Using a calculator, determine \( \log 5 \). The value is approximately 0.69897. This result will be used in the numerator of our initial expression.

Step 3: Calculate \( \log 1.07 \)

Using a calculator again, determine \( \log 1.07 \). The value is approximately 0.02938. This will be used in the denominator of our initial expression.

Step 4: Multiply \( 3 \log 1.07 \)

Multiply the result of \( \log 1.07 \) by 3: \( 3 \times 0.02938 = 0.08814 \). This value is the entire denominator of the original expression.

Step 5: Divide Results

Divide the result of \( \log 5 \) by the result of \( 3 \log 1.07 \): \( \frac{0.69897}{0.08814} \approx 7.93 \).

Step 6: Round Final Answer

Express the final result to the nearest hundredth as requested. The answer is approximately 7.93.

Key Concepts

Calculator Use in Algebra

Calculators in algebra are essential tools that assist in performing complex calculations like logarithms. In this exercise, we use a calculator to find the logarithmic values of numbers essential for solving the problem.

• To find the log of a number, simply enter the number into your calculator and press the "log" function.
• This function helps in quickly solving mathematical problems that require precision, such as our example with logarithms.

Using the calculator ensures accuracy, saving you time in manual calculations. It simplifies the process, reducing the likelihood of mistakes.
While calculators are a great aid, understanding the concepts behind the numbers is equally important for deeper comprehension.

Rounding Numbers

Rounding numbers is a crucial skill when dealing with decimals in mathematics. In our exercise, we are instructed to round the answer to the nearest hundredth, which is an essential step in giving our final result.

• Rounding helps in making numbers easier to work with or understand, especially in financial and statistical calculations.
• In this problem, the result, which is approximately 7.93, is an accurate representation to two decimal places.

Rounding involves looking at the third digit after the decimal point:

• If this digit is 5 or greater, you round up the second digit by one.
• If it's less than 5, the second digit remains the same.

Understanding rounding is a foundation for precise calculation, ensuring your answers are as accurate as possible.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. The expression we used, \( \frac{ \log 5}{3 \log 1.07} \), is an example of a more complex type, involving logarithms.

• The expression uses division, multiplication, and logarithms to solve a problem.
• Breaking down the expression into smaller parts makes it easier to comprehend and solve.

This is done by calculating each part like \( \log 5 \) and \( 3 \log 1.07 \) separately, then combining them through division. This step-by-step breakdown assists students in recognizing and understanding the components of algebraic expressions.

Division of Logarithms

Division of logarithms in algebra can seem challenging, but it becomes manageable through methodical steps. In our expression \( \frac{ \log 5}{3 \log 1.07} \), we divide two numbers that were derived using the logarithm operation.

• This technique involves finding both logarithmic values first and then applying basic arithmetic division.
• In this case, after computing \( \log 5 \) and \( 3 \log 1.07 \), you divide the two to find the quotient.

Division of logarithms can also relate to the subtraction of individual logarithm values when combined in advanced expressions:

• For instance, \( \log( \frac{a}{b}) = \log a - \log b \).

Such equations underscore the logical structure behind algebraic operations involving logarithms, making the subject both fascinating and systematic.