Question

Log10 10+log10 100+log10 1000+log10 10000+log10 100000

Answer 1

AnswEr :

Given Expression,

$\sf \: log_{10}(10) + log_{10}(100) + log_{10}(1000) + log_{10}(10000) + log_{10}(100000)$

Here,

• log(100) = log(10²)
• log(1000) = log(10³)
• log(10000) = log(10⁴)
• log(100000) = log(10^5)

Thus,

$\implies \sf \: log_{10}(10) + log_{10}( {10}^{2} ) + log_{10}(10 {}^{3} ) + log_{10}( {10}^{4} ) + log_{10}(10 {}^{5} )$

Using Power Rule of Logarithms,

$\implies \sf \: log_{10}(10) + 2 log_{10}(10) + 3 log_{10}(10) + 4 log_{10}(10) + 5 log_{10}(10) \\ \\ \implies \sf \: log_{10}(10) (1 + 2+ 3 + 4 + 5) \\ \\ \implies \sf \: 15 log_{10}(10)$

We know that,

log_{10}^10 = 1

Thus,

$\implies \sf \: 15 \times 1 \\ \\ \implies \sf \: 15$