Typical Logarithms and exponents!!

Algebra Level 2

$a, b, c$ are positive reals such that

$\frac {\log a}{b-c} = \frac {\log b}{c-a} = \frac {\log c}{a-b}.$

What is the value of

$a^a \times b^b \times c^c ?$

The answer is 1.000.

2 solutions

Saloni Arora
Dec 1, 2014

a loga / a(b-c) = b logb / b(c-a) = c*logc / c(a-b) = k (some value)

Sum of numerators/ Sum of denominators = k {If a/x= b/y= c/z = k, then a= kx, b= ky, c= kz Therefore, k= (a+b+c)/(x+y+z)}

So, we can say that a log a+b log b+ c log c= k (a(b-c) + b(c-a) + c(a-b)) a log a+b log b+ c log c= k (0) a log a+b log b+ c log c= 0 log a^a.b^b.c^c= 0

a^a.b^b.c^c= 1

Nice solution!

Ganeshkumar Ashokavardhanan - 6 years, 4 months ago

Akhil Vuriti
Nov 16, 2014

let a^a=k, b^b=k, c^c=k.. apply log principals and we goo

Why do you take all the three to be "k"? Could you elaborate?

Krishna Ar - 6 years, 6 months ago

You must instead take the equal expressions to be "K"

Krishna Ar - 6 years, 6 months ago

Downvoted!!

Daniel Arenson - 6 years, 6 months ago

Typical Logarithms and exponents!!

The answer is 1.000.

2 solutions

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