https://file.aiquickdraw.com/mathbot/file/1730240596201nkacd1q0.jpg

To solve each logarithmic expression to the nearest thousandth, we’ll use the change of base formula. The change of base formula for logarithms is:

\[
\log_b x = \frac{\log_{10} x}{\log_{10} b}
\]

Using this formula, let’s calculate each value.

### Problem Breakdown and Solution:

1. **\( \log_4 61 \):**

   \[
   \log_4 61 = \frac{\log_{10} 61}{\log_{10} 4} 
   \]
   Calculating:
   \[
   \log_{10} 61 \approx 1.785
   \]
   \[
   \log_{10} 4 \approx 0.602
   \]
   \[
   \log_4 61 = \frac{1.785}{0.602} \approx 2.966
   \]

2. **\( \log_2 26 \):**

   \[
   \log_2 26 = \frac{\log_{10} 26}{\log_{10} 2}
   \]
   Calculating:
   \[
   \log_{10} 26 \approx 1.415
   \]
   \[
   \log_{10} 2 \approx 0.301
   \]
   \[
   \log_2 26 = \frac{1.415}{0.301} \approx 4.702
   \]

3. **\( \log_2 60 \):**

   \[
   \log_2 60 = \frac{\log_{10} 60}{\log_{10} 2}
   \]
   Calculating:
   \[
   \log_{10} 60 \approx 1.778
   \]
   \[
   \log_{10} 2 \approx 0.301
   \]
   \[
   \log_2 60 = \frac{1.778}{0.301} \approx 5.906
   \]

4. **\( \log_4 49 \):**

   \[
   \log_4 49 = \frac{\log_{10} 49}{\log_{10} 4}
   \]
   Calculating:
   \[
   \log_{10} 49 \approx 1.690
   \]
   \[
   \log_{10} 4 \approx 0.602
   \]
   \[
   \log_4 49 = \frac{1.690}{0.602} \approx 2.808
   \]

### Final Answers (to the nearest thousandth):

1. \( \log_4 61 \approx 2.966 \)
2. \( \log_2 26 \approx 4.702 \)
3. \( \log_2 60 \approx 5.906 \)
4. \( \log_4 49 \approx 2.808 \)

Would you like a deeper explanation or have any questions?

#### Related Questions:
1. How does the change of base formula work for natural logarithms?
2. How can logarithmic expressions be simplified without a calculator?
3. Why do logarithmic values often have irrational results?
4. How do logarithmic functions behave when the base is less than 1?
5. What is the difference between logarithmic and exponential functions?

#### Tip:
Always check if you can simplify logarithmic expressions by recognizing powers, which can make calculations quicker without a calculator.

Use a calculator to approximate each logarithm to the nearest thousandth: 1) log₄ 61, 2) log₂ 26, 3) log₂ 60, 4) log₄ 49.

Learn how to approximate logarithmic expressions, including log₄ 61, log₂ 26, log₂ 60, and log₄ 49, to the nearest thousandth using the change of base formula. This solution is ideal for high school students looking to master logarithmic properties and calculator-based approximations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation