Question

If loge (a+b/2) =1/2( loge a+ loge b) then relation between a and b will be

Options
A) a=b
B)a = b/2
C) a = 2b
D) a= b/3

Please answer with solution

Answer 1

Answer:

A) a=b

Step-by-step explanation:

loge($\frac{a+b}{2}$)= $\frac{1}{2}$(loge a+ loge b)                                          

loge($\frac{a+b}{2}$)= $\frac{1\\}{2}$ (loge ab)                                              ( loga + logb= logab)

loge ($\frac{a+b}{2}$) = loge $ab^{1/2}$                                              ( x loga = log$a^{x}$)

  $\frac{a+b}{2}$ = $ab^{1/2}$                                                                 ( loga= logb; i.e. a=b)

(now squaring both side)

$\frac{a^{2} +b^{2} + 2ab}{2} =ab$

$a^{2} +b^{2}+2ab = 4ab$

$a^{2} +b^{2}+2ab-4ab= 0$

$a^{2} +b^{2}-2ab =0$

$(a-b)^{2} =0$

which is only true at a=b.