Problem 21 - Algebra 2 and Trigonometry

Question

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \frac{3 \log 4}{\log 5} $$

Step-by-Step Solution

Answer

The value of \( \frac{3 \log 4}{\log 5} \) is approximately 2.57.

Step 1: Write the Given Expression

The expression given is \( \frac{3 \log 4}{\log 5} \). Our task is to evaluate this expression.

Step 2: Use a Calculator for Logarithms

Find the logarithm values using a calculator. First, calculate \( \log 4 \) and \( \log 5 \). To the nearest hundredth:\[ \log 4 \approx 0.60 \]\[ \log 5 \approx 0.70 \]

Step 3: Multiply and Simplify Numerator

Multiply \( \log 4 \) by 3 to get the numerator's value:\[ 3 \times \log 4 = 3 \times 0.60 = 1.80 \]

Step 4: Divide for the Final Result

Divide the result from Step 3 by \( \log 5 \):\[ \frac{1.80}{0.70} \approx 2.57 \]

Step 5: Round Result to Nearest Hundredth

The calculated result, \( 2.57 \), is already rounded to the nearest hundredth.

Key Concepts

Understanding Step-by-Step Calculation

Calculating logarithmic expressions might initially seem challenging, but it's easier when broken down into steps.
By using a step-by-step approach, you get a clear path to follow and can handle each part of the expression carefully.
Here's a brief rundown of the process:

• Identify the given expression and understand what is being asked.
• Utilize a calculator to find necessary logarithmic values, ensuring they are accurate to necessary precision, such as the nearest hundredth.
• Multiply or adjust components as needed before combining.
• Perform division or any other operations to simplify and finalize the expression.

This method makes sure no part of the calculation is overlooked. With practice, this systematic approach makes evaluating logarithmic expressions much more straightforward, reducing errors and increasing confidence.

Working with Logarithmic Expressions

Logarithms are the inverse operation to exponentiation, and they frequently appear in math problems that involve exponential growth or decay.
In the expression \( \frac{3 \log 4}{\log 5} \), each part has a specific role:

• \( \log 4 \) and \( \log 5 \): These terms represent the logarithms of 4 and 5, respectively. They indicate the power to which a base number (usually 10 or e) must be raised to produce those numbers.
• Multiplication: The expression involves multiplying \( \log 4 \) by 3. This multiplication affects only the numerator and is essential for adjusting the expression to obtain the correct result.
• Division: Once the numerator is computed, it is divided by \( \log 5 \). This final division step combines the terms, yielding the final evaluated expression.

The beauty of logarithmic expressions lies in their ability to simplify complex calculations, helping you find solutions to exponential equations.

Approximating Values Confidently

Approximations are crucial in many mathematical situations, especially when dealing with logarithmic calculations where exact values might not arise easily.
When using a calculator for logarithms, you aim to get a close estimate that is sufficiently accurate for your needs, typically to a specific decimal point like the nearest hundredth.
In evaluating the expression \( \frac{3 \log 4}{\log 5} \), the following approximations were used:

• \( \log 4 \approx 0.60 \): The value is rounded to the nearest hundredth to simplify further calculations.
• \( \log 5 \approx 0.70 \): Similarly, this value is rounded for precision without sacrificing accuracy.

After applying these approximations in the multiplication and division steps, you ensure the final result, 2.57, remains both accurate and appropriately precise.
Learning to comfortably approximate values not only aids in manual calculations but also strengthens understanding of when and why rounding is acceptable.