Short Answer 
The change of base formula allows calculation of logarithms in different bases, expressed as log_b(x) = log(x) / log(b). Using this formula, we can simplify log 10/log 2 * log 8/log 4 * log 4/log 10 to log 8/log 2, which further simplifies to 3, confirming that log 8/log 2 equals 3.
Step 1: Understand the Change of Base Formula
The change of base formula allows us to calculate logarithms in different bases. It states that for any positive numbers (b), (x), and (y) (where (b neq 1)), the equation can be written as:
- log_b(x) = log(x) / log(b)
- This means you can convert the logarithm to use a new base for easier calculations.
- In our case, we will convert various logarithmic terms using this formula.
Step 2: Rewrite the Expression Using the Formula
Using the change of base formula, we can rewrite the original problem as follows:
- Combine all logarithmic expressions into one equation: log 10/log 2 * log 8/log 4 * log 4/log 10.
- This simplifies to log 8/log 2 when similar terms (log 10) cancel out.
- Now, we need to show that log 8/log 2 equals 3 by further simplifying.
Step 3: Simplify and Solve
We want to evaluate log 8/log 2, and we can simplify it as follows:
- log 8 can be rewritten as log 2^3 since 8 is 2 raised to the power of 3.
- This allows us to express the logarithm in terms of base 2: log 2^3/log 2^1.
- This simplifies to (3 log 2)/(1 log 2), resulting in the final answer of 3.