Log

Algebra Level 3

log ⁡ 10 ( 1 ) + log ⁡ 10 ( 10 ) + log ⁡ 10 ( 100 ) + ⋯ + log ⁡ 10 ( 10 000 000 000 ) = ? \log_{10} (1 ) + \log_{10} ({10}) + \log_{10} ({100} ) + \cdots + \log_{10} ({\num{10000000000}}) = \, ?


The answer is 55.

Dhruv Sharma by Dhruv Sharma , 19, India

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1 solution

log ⁡ ( 1 0 10 × 11 2 ) log ⁡ ( 10 ) = log ⁡ ( 1 0 55 ) log ⁡ ( 10 ) = 55 \Huge{\dfrac{\log(10^{\frac{10\times 11}{2}})}{\log(10)}=\dfrac{\log(10^{55})}{\log(10)}=\boxed{55}}

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