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We are asked to evaluate the logarithmic expression \(\log_{1000} 10\).

The logarithmic expression \(\log_b a = x\) means that \(b^x = a\). In this case, we have:

\[
\log_{1000} 10 = x \quad \text{which means} \quad 1000^x = 10.
\]

We can rewrite \(1000\) and \(10\) as powers of 10:
\[
1000 = 10^3 \quad \text{and} \quad 10 = 10^1.
\]

So we can substitute this into the equation:
\[
(10^3)^x = 10^1.
\]

Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we get:
\[
10^{3x} = 10^1.
\]

Since the bases are the same, we can equate the exponents:
\[
3x = 1.
\]

Solving for \(x\):
\[
x = \frac{1}{3}.
\]

Thus, \(\log_{1000} 10 = \frac{1}{3}\).

Would you like further details or have any questions?

Here are five related questions you might find interesting:

1. How would you evaluate \(\log_{1000} 100\)?
2. What is the value of \(\log_2 16\)?
3. How do you simplify logarithmic expressions with different bases, such as \(\log_3 9\)?
4. Can you explain the change of base formula for logarithms?
5. What is the relationship between logarithms and exponential functions?

**Tip:** When solving logarithmic equations, it's useful to rewrite the base and the argument in terms of powers of the same number if possible. This simplifies the equation!

Evaluate. Write your answer as a whole number, proper fraction, or improper fraction in simplest form. log_1000 10 = ?

Learn how to evaluate log base 1000 of 10 as a fraction. This problem covers the concepts of logarithms and exponents, using power rules to simplify the equation. Suitable for students in Grades 9-12.

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