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.