To solve the expression \((\log_5 16)(\log_{16} 5)\), we can use the change of base formula and a known identity in logarithms.

### Change of Base Formula:
\[
\log_a b = \frac{\log_c b}{\log_c a}
\]
for any base \( c \).

### Identity:
\[
(\log_a b)(\log_b a) = 1
\]
This identity holds for any positive numbers \( a \) and \( b \), where \( a \neq 1 \) and \( b \neq 1 \).

Given expression:
\[
(\log_5 16)(\log_{16} 5)
\]

Applying the identity:
\[
(\log_5 16)(\log_{16} 5) = 1
\]

### Final Answer:
\[
\boxed{1}
\]

Would you like further details or have any questions?

Here are some related questions:

1. How is the change of base formula derived?
2. What other logarithmic identities are commonly used in algebra?
3. How can you express \(\log_5 16\) in terms of natural logarithms?
4. What are the conditions under which the identity \((\log_a b)(\log_b a) = 1\) is valid?
5. How can this identity be used to simplify logarithmic expressions in general?

**Tip:** Understanding the basic properties of logarithms, such as the product, quotient, and power rules, is key to simplifying and solving logarithmic equations efficiently.

Discover how to solve the logarithmic expression (log5 16)(log16 5) using the change of base formula and logarithmic identities. Learn key concepts like the change of base formula and the identity (log_a b)(log_b a) = 1. This problem is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation