Log base 5 6 =a. B= log base 5 11. Use change of base formula. Rewrite each expression in terms of a and b

We are given:

- \( \log_5 6 = a \)
- \( \log_5 11 = b \)

We need to rewrite these logarithms using the change of base formula and express them in terms of \( a \) and \( b \).

### Change of Base Formula:
The change of base formula states that:

\[
\log_b x = \frac{\log_c x}{\log_c b}
\]

Where \( c \) is any base (typically base 10 or \( e \) for natural logs). However, in this case, we can use the known values of \( a \) and \( b \) to simplify expressions involving logarithms in base 5.

### Expressions to Rewrite:

1. \( \log_5 66 \)
   - By the property of logarithms:
   
     \[
     \log_5 66 = \log_5 (6 \times 11) = \log_5 6 + \log_5 11
     \]
     Using the definitions of \( a \) and \( b \):
   
     \[
     \log_5 66 = a + b
     \]

2. \( \log_5 \left( \frac{11}{6} \right) \)
   - Using the quotient property of logarithms:
   
     \[
     \log_5 \left( \frac{11}{6} \right) = \log_5 11 - \log_5 6
     \]
     Which simplifies to:
   
     \[
     \log_5 \left( \frac{11}{6} \right) = b - a
     \]

### Final Answer:
- \( \log_5 66 = a + b \)
- \( \log_5 \left( \frac{11}{6} \right) = b - a \)

Would you like to explore more details or have any other questions?

Here are 5 related questions for practice:
1. What is the value of \( \log_5 36 \) in terms of \( a \)?
2. How can you express \( \log_5 1 \) using properties of logarithms?
3. Use the change of base formula to express \( \log_2 6 \) in terms of \( \log_5 6 \) and \( \log_5 2 \).
4. How do you express \( \log_5 (6^2) \) in terms of \( a \)?
5. How do you express \( \log_5 \left( \frac{6^2}{11} \right) \) in terms of \( a \) and \( b \)?

Tip: Logarithm properties like the product, quotient, and power rules are extremely useful for simplifying expressions.

Log base 5 6 = a. B= log base 5 11. Use change of base formula. Rewrite each expression in terms of a and b.

Learn how to use the change of base formula and properties of logarithms to express log base 5 of 6 and log base 5 of 11 in terms of a and b. This detailed solution breaks down the product and quotient properties of logarithms. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation