Question

Prove that log32 log43 log5 4...log1514 log1615 =
1
4​

Answer 1

Answer:

sorry I don't know as I'm not learnt that

Answer 2

Answer:

Step-by-step explanation:

To prove,

log₃2 ×log₄3× log₅4× ..........×log₁₅14 ×log₁₆15 =$\frac{1}{4}$

Recall the formula

Change of base rule of logarithm

logₐ b= $\frac{log\ b}{log\ a}$

Power rule of logarithm

log aⁿ = n×log a

Solution

LHS = log₃2 ×log₄3× log₅4× ..........×log₁₅14 ×log₁₆15

Applying the change of base rule of logarithm we get

log₃2 ×log₄3× log₅4× ..........×log₁₅14 ×log₁₆15

= $\frac{log\ 2}{log\ 3} X \frac{log \ 3}{log \ 4} X\frac{log \ 4}{log \ 5} X...............\frac{log\ 14}{log \ 15} X \frac{log\ 15}{log \ 16}$

canceling the terms we get

log₃2 ×log₄3× log₅4× ..........×log₁₅14 ×log₁₆15

= $\frac{log\ 2}{log \ 16}$

= $\frac{log\ 2}{log \ 2^4}$

Applying the power rule of logarithm we get

= $\frac{log\ 2}{4log\ 2}$

=$\frac{1}{4}$

∴ log₃2 ×log₄3× log₅4× ..........×log₁₅14 ×log₁₆15 =$\frac{1}{4}$

Hence proved

SPJ3