Question

16.If ab = bc = ab, then b + c always equals(1)1bc(2)12bc(3)1(

Answer 1

question is -----> if a^b = b^c = ab. then b + c always equals (1) bc, (2) 12bc (3) b - c (4) 1/4 bc

Given, $a^b=b^c=ab$

$a^b=ab$

taking log both sides, 

$loga^b=logab$

we know, $logAB=logA+logB$
$logA^B=BlogA$

so, $bloga=loga+logb$

$bloga-loga=logb$

$(b-1)loga=logb$

$loga^{(b-1)}=logb$

removing log from both sides, 

$a^{(b-1)}=b$

similarly, [/tex]b^c=ab[/tex]

put value of b , 

$(a^{(b-1)})^c=aa^{(b-1)}$

$a^{cb-c}=a^{(b-1)+1}$

bc - c = bbc−c=b 

b + c = bc 

hence, b + c = bc

Answer 2

a^b= b^c = ab

a^b = ab

Now we will take the log on both sides.

log a^b = logab

we know that log AB = log A + log B

logA^B = BlogA

bloga = loga+logb

bloga - loga = logb

(b-1) loga = logb

log^(b-1) = logb

Now we will remove the log from both sides.

a^(b-1) = b

now we will put the value of b

(a^(b-1))^c = aa^(b-1)

a^cb-c = a^(b-1)+1

bc - c = bbc−c=b  

b + c = bc  

hence, b + c = bc

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