How would you show \(\displaystyle{{\log}_{{2}}{3}}+{{\log}_{{3}}{4}}+{{\log}_{{4}}{5}}+{{\log}_{{5}}{6}}{>}{5}?\) After trying to represent

Jasmine Todd

Jasmine Todd

Answered question

2022-03-30

How would you show
log23+log34+log45+log56>5?
After trying to represent the mentioned expression in the same base, a messy expression is created. The hint mentioned that the proof can take help of quadratic equations.
Could you provide some input? Is it a good idea to think of some graphical solution?

Answer & Explanation

Esteban Sloan

Esteban Sloan

Beginner2022-03-31Added 21 answers

log23+log34+log45+log56>4×log3log2×log4log3×log5log4×log6log54=4×log264.
Now, 54=1.25. So we are done if we can show that 1.254<2.45<log26. To see this, observe that 22.5=25=32<6. That gives you the result.

shvatismop1rj

shvatismop1rj

Beginner2022-04-01Added 10 answers

log23>1.58350>279(38213)6>29,
log34>1.26450>363(4739)7>14,
and log45>1.16525>429(5546)5>14.
It follows that
log23+log34+log45>1.58+1.26+1.16=4.

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