Evaluate the expression. (Simplify your answer completely.)
(a)    log3(3)


(b)    log9(1)


(c)    log6(65)

Let's evaluate each logarithmic expression step by step.

---

**(a) \( \log_3(3) \):**

- The logarithmic expression \( \log_b(x) \) asks the question: "To what power must the base \( b \) be raised to get \( x \)?"
- In this case, \( \log_3(3) \) means: "To what power must 3 be raised to get 3?"
- The answer is \( 1 \), since \( 3^1 = 3 \).

**So, \( \log_3(3) = 1 \).**

---

**(b) \( \log_9(1) \):**

- Any logarithm of 1 is 0, regardless of the base. This is because any non-zero number raised to the power of 0 is 1.
- Therefore, \( \log_9(1) = 0 \), since \( 9^0 = 1 \).

**So, \( \log_9(1) = 0 \).**

---

**(c) \( \log_6(65) \):**

- \( \log_6(65) \) does not simplify to an exact integer since 65 is not a power of 6.
- To express it in a different way or estimate it, we can use the change of base formula:
  \[
  \log_b(x) = \frac{\log(x)}{\log(b)}
  \]
- Using this formula:
  \[
  \log_6(65) = \frac{\log(65)}{\log(6)}
  \]
  This would give a decimal approximation, but it's the exact way to express it algebraically.

---

### Summary:
- \( \log_3(3) = 1 \)
- \( \log_9(1) = 0 \)
- \( \log_6(65) = \frac{\log(65)}{\log(6)} \)

Would you like me to calculate the decimal approximation for part (c) or give more details on logarithmic properties?

---

### Related Questions:
1. What is \( \log_5(25) \)?
2. How do you simplify \( \log_7(49) \)?
3. What is the general formula for changing the base of a logarithm?
4. How do you solve logarithmic equations like \( \log_2(x) = 3 \)?
5. What is the value of \( \log_{10}(1000) \)?

### Tip:
The logarithm of 1 in any base is always 0, since \( b^0 = 1 \) for any base \( b \).

Evaluate the expression. (Simplify your answer completely.) (a) log3(3), (b) log9(1), (c) log6(65)

Learn how to evaluate logarithmic expressions like log3(3), log9(1), and log6(65. This detailed guide explains logarithmic properties, including the base rule, exponentiation, and the change of base formula. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation