$\log_{10} (10 \times 10^2 \times 10^3 \times ...... \times 10^9)$ is

Concept & Formula This problem combines the product rule of exponents with the fundamental property of base-10 logarithms. When multiplying expressions with the same base, you add their exponents: $$a^m \times a^n = a^{m+n}$$ Additionally, the logarithm property states: $$\log_{10} (10^k) = k$$ Step-by-Step Solution * **Given:** $\log_{10} (10 \times 10^2 \times 10^3 \times \dots \times 10^9)$ * **Simplify the Argument:** Use the exponent rule to combine the terms inside the parenthesis. The base is 10 for all terms, so we sum the exponents from 1 to 9. $$10^{1 + 2 + 3 + \dots + 9}$$ * **Calculate the Sum:** Use the formula for the sum of the first $n$ natural numbers, which is $\frac{n(n+1)}{2}$, where $n = 9$. $$\text{Sum} = \frac{9(9+1)}{2} = \frac{9 \times 10}{2} = 45$$ * **Substitute the Sum:** The argument simplifies to $10^{45}$. * **Evaluate the Logarithm:** Substitute this back into the original logarithmic expression. $$\log_{10} (10^{45})$$ * **Final Calculation:** Applying the rule $\log_a (a^x) = x$, the log and the base 10 cancel each other out. $$45$$ Exam Strategy & Shortcut Recognize the pattern immediately: this is just the sum of numbers from 1 to 9. Knowing that the sum of 1 through 9 is 45 (a common arithmetic fact) allows you to bypass writing out any steps. The question instantly reduces to $\log_{10}(10^{45}) = 45$ in your head, taking less than 5 seconds. Common Pitfall A common error is confusing the product rule of exponents with multiplication, leading students to multiply the exponents (e.g., $1 \times 2 \times 3 \dots \times 9 = 9!$) instead of adding them. Remember that multiplying same bases means adding their powers. Final Answer **Therefore, the correct answer is 45.**