Question

show that log 360 =3log2+2log3+log5

Answer 1

360=2×2×2×3×3×5=2cube×3 square×5
log 360=log (2cube×3square×5)
log 360=log 2cube+log 3square+log 5
log360= 3log 2+2log3+log5

Answer 2

Given:  

log 360 =3log2 + 2log3 + log5

To prove:  

LHS = RHS  

Solution:  

Taking right hand side from the given  

3log2 + 2log3 + log5

To solve this, we have to apply some of the logarithmic rules  

$a \log b=\log b^{a}$

$\log a b=\log a+\log b$

Applying the above given rules  

$=3 \log 2+2 \log 3+\log 5$

Applying the first rule given then it become

$=\log 2^{3}+\log 3^{2}+\log 5$

$=\log 8+\log 9+\log 5$

Applying the second rule given then it become

$=\log (8 \times 9 \times 5)$

=log360 = RHS  

Therefore, LHS = RHS  

3log2 + 2log3 + log5 = log360  

Hence proved.