Question

Find the value of \(\log_{10}10+\log_{10}10^2+\log_{10}10^3+\ldots+\log_{10}10^n\)

• A. \(n^2+1\)
• B. \(n^2-1\)
• C. \(\dfrac{n(n+1)}{3}\)
• D. \(\dfrac{n^2+n}{2}\)

Hint:

Each log₁₀(10^k) simplifies to just k, turning the sum into 1+2+…+n.

Answer 1

The Correct Option is D

Step 1: Simplify each term.
Using the log rule \(\log_a a^k=k\), each term \(\log_{10}10^k=k\).

Step 2: Rewrite the sum.
The expression becomes \(1+2+3+\ldots+n\).

Step 3: Apply the sum of the first n natural numbers.
\(1+2+\ldots+n=\dfrac{n(n+1)}{2}=\dfrac{n^2+n}{2}\).

Step 4: Final Answer.
The value is \(\dfrac{n^2+n}{2}\), so option D is correct.