https://r2.erweima.ai/mathbot/image/1726079097283uipdhrgn.jpg

The equation you have shared is:

\[
\log_6{x} - \log_{216}{x} = \log_6{4}
\]

### Step-by-step Solution:

1. **Change of Base for \(\log_{216}{x}\)**:
   We can rewrite \(\log_{216}{x}\) using the change of base formula:
   \[
   \log_{216}{x} = \frac{\log_6{x}}{\log_6{216}}
   \]
   Now, we know that \( 216 = 6^3 \), so:
   \[
   \log_6{216} = 3
   \]
   Therefore:
   \[
   \log_{216}{x} = \frac{\log_6{x}}{3}
   \]

2. **Substitute into the equation**:
   Now substitute \(\log_{216}{x}\) back into the original equation:
   \[
   \log_6{x} - \frac{\log_6{x}}{3} = \log_6{4}
   \]

3. **Combine like terms**:
   Factor out \(\log_6{x}\) on the left-hand side:
   \[
   \log_6{x} \left( 1 - \frac{1}{3} \right) = \log_6{4}
   \]
   Simplify the expression inside the parentheses:
   \[
   \log_6{x} \left( \frac{2}{3} \right) = \log_6{4}
   \]

4. **Solve for \(\log_6{x}\)**:
   Multiply both sides by \(\frac{3}{2}\) to isolate \(\log_6{x}\):
   \[
   \log_6{x} = \frac{3}{2} \log_6{4}
   \]

5. **Simplify the right-hand side**:
   Using the logarithmic property \(\log_b{a^k} = k \log_b{a}\), we know that \(\log_6{4} = \log_6{2^2} = 2 \log_6{2}\). So:
   \[
   \log_6{x} = \frac{3}{2} \times 2 \log_6{2} = 3 \log_6{2}
   \]

6. **Exponentiate both sides**:
   Finally, raise both sides as powers of 6 to eliminate the logarithms:
   \[
   x = 6^{3 \log_6{2}} = 2^3 = 8
   \]

### Solution:
\[
x = 8
\]

Let me know if you'd like further details or clarifications! 

Here are 5 related questions:
1. How do you use the change of base formula for logarithms?
2. Why is \( \log_6{216} = 3 \) in this problem?
3. Can you solve this equation using logarithmic properties without the change of base formula?
4. What other strategies could be used to simplify logarithmic expressions?
5. How would the solution change if the logarithm bases were different?

**Tip:** When solving logarithmic equations, always try to simplify logarithms with different bases to a common base if possible.

Learn how to solve the logarithmic equation log_6(x) - log_216(x) = log_6(4) using the change of base formula and logarithmic properties. This detailed solution breaks down each step to help you understand how to manipulate logarithmic expressions and solve for x.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation