Question

l og 1/2 + log 2/3 + log 3/4 + log 4) 5 - log 1/5 isequal to 9
​

Answer 1

Given log (1/2)  + log (2/3)  + log (3/4)  + log (4/5)  - log (1/5)

To Find : Value

Solution:

log (a/b) = log a - log b

log (1/2) = log 1 - log 2

log (2/3) = log 2 - log 3

log (3/4) = log 3 - log 4

log (4/5) = log 4 - log 5

log (1/5) = log 1 - log 5

log (1/2)  + log (2/3)  + log (3/4)  + log (4/5)  - log (1/5)

= log 1 - log 2 +  log 2 - log 3 + log 3 - log 4  + log 4 - log 5 - ( log 1 - log 5)

=  log 1   - log 5 - ( log 1 - log 5)

= log1  - log 5 - log 1 + log 5

= log1  - log 1 - log 5 + log 5

= 0

log 1/2  + log 2/3  + log 3/4  + log 4/5  - log 1/5  = 0

Learn More:

if x=log [a-b]; y=log [a+b] ; z= log [a2 - Brainly.in

https://brainly.in/question/13214148

If x=log(a+b) y=log(a-b) z=log(a²-b²). Then what is the relation ...

https://brainly.in/question/13203506

Answer 2

SOLUTION

TO PROVE

$\displaystyle \sf{ \log \bigg( \frac{1}{2} \bigg) +\log \bigg( \frac{2}{3} \bigg) + \log \bigg( \frac{3}{4} \bigg) + \log \bigg( \frac{4}{5} \bigg) - \log \bigg( \frac{1}{5} \bigg) = 0}$

FORMULA TO BE IMPLEMENTED

We are aware of the formula on logarithm that

$\displaystyle \sf{ \log m + \log n = \log (mn)}$

EVALUATION

$\displaystyle \sf{ \log \bigg( \frac{1}{2} \bigg) +\log \bigg( \frac{2}{3} \bigg) + \log \bigg( \frac{3}{4} \bigg) + \log \bigg( \frac{4}{5} \bigg) - \log \bigg( \frac{1}{5} \bigg) }$

$\displaystyle \sf{ = \log \bigg( \frac{1}{2} \times \frac{2}{3} \bigg) + \log \bigg( \frac{3}{4} \bigg) + \log \bigg( \frac{4}{5} \bigg) - \log \bigg( \frac{1}{5} \bigg) } \: ( \: by \: above \: formula \: )$

$\displaystyle \sf{ = \log \bigg( \frac{1}{3} \bigg) + \log \bigg( \frac{3}{4} \bigg) + \log \bigg( \frac{4}{5} \bigg) - \log \bigg( \frac{1}{5} \bigg) } \:$

$\displaystyle \sf{ = \log \bigg( \frac{1}{3} \times \frac{3}{4} \bigg) + \log \bigg( \frac{4}{5} \bigg) - \log \bigg( \frac{1}{5} \bigg) } \: ( \: by \: above \: formula \: )$

$\displaystyle \sf{ = \log \bigg( \frac{1}{4} \bigg) + \log \bigg( \frac{4}{5} \bigg) - \log \bigg( \frac{1}{5} \bigg) } \:$

$\displaystyle \sf{ = \log \bigg( \frac{1}{4} \times \frac{4}{5} \bigg) - \log \bigg( \frac{1}{5} \bigg) } \: ( \: by \: above \: formula \: )$

$\displaystyle \sf{ = \log \bigg( \frac{1}{5} \bigg) - \log \bigg( \frac{1}{5} \bigg) } \:$

$= 0$

Hence proved

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. if 2log x base y=6,then the relation between x and y

https://brainly.in/question/27199002

2. If 2log((x+y)/(4))=log x+log y then find the value of (x)/(y)+(y)/(x)

https://brainly.in/question/24081206