https://file.aiquickdraw.com/mathbot/file/1732111859529m79aos10.jpg

To solve the given problem \( \log_8 32 + \log_8 2 \), let's proceed step by step:

### Step 1: Use logarithmic addition property
The addition property of logarithms states:
\[
\log_a b + \log_a c = \log_a (b \cdot c)
\]
So:
\[
\log_8 32 + \log_8 2 = \log_8 (32 \cdot 2)
\]

### Step 2: Simplify the argument inside the logarithm
\[
32 \cdot 2 = 64
\]
Thus, the expression becomes:
\[
\log_8 64
\]

### Step 3: Evaluate \( \log_8 64 \)
We need to find the value of \( x \) such that:
\[
8^x = 64
\]
Since \( 8 = 2^3 \) and \( 64 = 2^6 \), we can rewrite \( 8^x = 64 \) as:
\[
(2^3)^x = 2^6
\]
\[
2^{3x} = 2^6
\]
Equating the exponents:
\[
3x = 6 \quad \Rightarrow \quad x = 2
\]

### Final Answer:
\[
\log_8 32 + \log_8 2 = 2
\]

Would you like further clarification or a detailed explanation of any step?  

### 5 Related Questions:
1. How do logarithmic properties simplify complex calculations?
2. Can we generalize the steps for solving \( \log_a (b) + \log_a (c) \)?
3. How do changes in the base of a logarithm affect the calculations?
4. What is the importance of converting numbers to the same base in logarithmic equations?
5. Can you solve \( \log_8 16 + \log_8 4 \) using similar steps?

### One Tip:
Always remember that logarithms are the inverse operations of exponentiation, which makes them useful for solving exponential equations effectively!

Find the value of the equation log₈ 32 + log₈ 2.

Step-by-step solution to the logarithmic equation log₈ 32 + log₈ 2. This example demonstrates the use of logarithmic addition properties and exponentiation. Learn how to simplify and evaluate logarithms effectively. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation