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The given equation is:

\[
\log_2 3 \times \log_3 4 \times \log_4 5 \times \cdots \times \log_x (x+1) = 10
\]

### Step-by-Step Solution:

This type of equation involves a product of logarithms with bases and arguments progressing incrementally. We can utilize the change of base formula for logarithms to simplify the expression.

#### Step 1: Recognize the general pattern.
Each term in the product can be simplified using the change of base formula:
\[
\log_b a = \frac{\log a}{\log b}
\]
This allows us to rewrite each term in the product as:
\[
\log_2 3 = \frac{\log 3}{\log 2}, \quad \log_3 4 = \frac{\log 4}{\log 3}, \quad \log_4 5 = \frac{\log 5}{\log 4}, \quad \dots, \quad \log_x (x+1) = \frac{\log(x+1)}{\log x}
\]

#### Step 2: General simplification.
The product of these logarithms telescopes. This is because for each adjacent logarithm, the numerator of one term is canceled by the denominator of the previous term:
\[
\frac{\log 3}{\log 2} \times \frac{\log 4}{\log 3} \times \frac{\log 5}{\log 4} \times \cdots \times \frac{\log(x+1)}{\log x} = \frac{\log(x+1)}{\log 2}
\]

#### Step 3: Set up the equation.
Now, the equation reduces to:
\[
\frac{\log(x+1)}{\log 2} = 10
\]

#### Step 4: Solve for \( x \).
Multiply both sides by \( \log 2 \) to isolate the logarithm:
\[
\log(x+1) = 10 \times \log 2
\]
\[
\log(x+1) = \log(2^{10})
\]
\[
\log(x+1) = \log 1024
\]

Since the logarithms are equal, their arguments must be equal:
\[
x + 1 = 1024
\]
\[
x = 1023
\]

Thus, the solution to the equation is \( x = 1023 \).

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Let me know if you want more details or have any questions.

Here are 5 related questions for further exploration:
1. How does the change of base formula work in logarithms?
2. What is a telescoping product and how does it simplify this type of equation?
3. Can this problem be generalized to different logarithmic products?
4. What would happen if the product had a finite or infinite number of terms?
5. What is the role of exponential growth in logarithmic equations?

**Tip:** The change of base formula is a crucial tool when dealing with logarithms of different bases. Always check if adjacent terms can telescope for simplification.

Solve the equation log_2 3 * log_3 4 * log_4 5 * ... * log_x (x+1) = 10.

Step-by-step solution to solve a complex logarithmic equation involving change of base formula and telescoping product. Learn how to simplify and solve log_2 3 * log_3 4 * ... * log_x (x+1) = 10.

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