https://file.aiquickdraw.com/mathbot/file/1725812856294cqdyw4kd.jpg

The expression shown in the image is:

\[
2 \log 16 - 3 \log \left( \frac{1}{27} \right) + 5 \log 1
\]

Let's break this down and simplify it step by step.

1. **Simplify \( \log 16 \)**:
   \[
   \log 16 = \log (2^4) = 4 \log 2
   \]
   So, 
   \[
   2 \log 16 = 2 \times 4 \log 2 = 8 \log 2
   \]

2. **Simplify \( \log \left( \frac{1}{27} \right) \)**:
   \[
   \log \left( \frac{1}{27} \right) = \log (27^{-1}) = -\log 27
   \]
   Since \( 27 = 3^3 \), we can write:
   \[
   \log 27 = \log (3^3) = 3 \log 3
   \]
   So,
   \[
   -\log 27 = -3 \log 3
   \]
   and
   \[
   3 \log \left( \frac{1}{27} \right) = 3 \times (-3 \log 3) = -9 \log 3
   \]

3. **Simplify \( 5 \log 1 \)**:
   \[
   \log 1 = 0
   \]
   Therefore,
   \[
   5 \log 1 = 5 \times 0 = 0
   \]

Now, putting everything together:
\[
8 \log 2 - 9 \log 3 + 0
\]

This is the simplified form of the expression:
\[
8 \log 2 - 9 \log 3
\]

Would you like more details or have any further questions?

Here are some related questions for practice:
1. How would you simplify \( 3 \log 8 + 2 \log 5 \)?
2. What happens when you multiply two logarithms, for example, \( \log a \times \log b \)?
3. Can you simplify \( \log (2 \times 8) \)?
4. How does the logarithmic identity \( \log (a^b) = b \log a \) work in other examples?
5. How do you simplify \( \log \frac{1}{x} \)?

**Tip**: Remember that \( \log 1 = 0 \), which can often simplify expressions.

This guide provides a detailed step-by-step solution to the logarithmic expression 2log(16) - 3log(1/27) + 5log(1). Learn how to simplify logarithms using the power rule and logarithm of 1. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation