To solve the expression \( ^5\log(\sqrt{11} + 6) + ^5\log(\sqrt{11} - 6) \), we can use the logarithm property that states:

\[
\log_b(x) + \log_b(y) = \log_b(xy)
\]

In this case, \( b = 5 \), \( x = \sqrt{11} + 6 \), and \( y = \sqrt{11} - 6 \).

So, we can combine the logarithms as follows:

\[
^5\log(\sqrt{11} + 6) + ^5\log(\sqrt{11} - 6) = ^5\log\left[(\sqrt{11} + 6)(\sqrt{11} - 6)\right]
\]

Now, we simplify the product using the difference of squares formula:

\[
(\sqrt{11} + 6)(\sqrt{11} - 6) = (\sqrt{11})^2 - 6^2 = 11 - 36 = -25
\]

Thus, the expression becomes:

\[
^5\log(-25)
\]

However, logarithms of negative numbers are undefined in the real number system. Therefore, the expression \( ^5\log(-25) \) is undefined in the real numbers.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What are the conditions under which logarithms are defined?
2. How does the difference of squares formula work?
3. Can logarithms of negative numbers be defined in the complex plane?
4. What are some other properties of logarithms?
5. How do we handle logarithms with different bases?

**Tip:** Always check if the arguments of a logarithm are within the domain of real numbers to ensure the expression is defined.

Explore the step-by-step solution to the logarithmic expression ⁵log(√11+6) + ⁵log(√11-6). This problem highlights the use of logarithm properties and the difference of squares formula, 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