If x>=y and y>1 ,then value of the expression log x(in the base)(x/y) + log y(in the base)(y/x) can never be

Question

If x>=y and y>1 ,then value of the expression log x(in the base) (x/y) + log y(in the base) (y/x) can never be
(a) -1
(b) -0.5
(c) 0
(d) 1

Answer 1

I'm a little confused in your notation - when you say log x(in the base)(x/y), do you mean the logarithm base x/y of x, or the logarithm base x of x/y? I shall figure the problem for both.

In case 1:

log x/log (x/y) + log y/log (y/x)
log x/(log x - log y) + log y/(log y - log x)
log x/(log x - log y) - log y/(log x - log y)
(log x - log y)/(log x - log y)
1

In this case, the value of your expression can never be -1, -0.5, 0, or indeed anything other than 1.

In case 2:

log (x/y)/log x + log (y/x)/log y
(log x - log y)/log x + (log y - log x)/log y
1 - log y/log x + 1 - log x/log y
2-(log y/log x + log x/log y)
note that ∀n, |n+1/n|≥2
≤2-2
≤0

In which case, the value of your expression can never be 1

Answer 2

Few regulations to undergo in concepts whilst condensing logarithms: a log b = log b^a - skill Rule log b + log a = log ba = log (b*a) - Addition Rule log b - log a = log (b/a) - Subtraction Rule 2 log x + 4 log y - 2 log z skill Rule: log x^2 + log y^4 - log z^2 Addition Rule for the 1st 2 words: log (x^2 * y^2) - log z^2 Subtraction Rule: log ((x^2-y^2)/z^2)

Answer 3

it can never be positive, so the fourth answer...

question simplifies to 2-(M+1/M)
M+1/M greater than or equal to 2 using am>GM

Answer 4

log(x)(x/y) + log(y)(y/x)
(log(x)(x) - log(x)y) + (log(y)(y) - log(y)(x))
(1 - log(x)y) + (1 - log(y)(x))

1 - log(x)y + 1 - log(y)(x)
2 - log(x)y - log(y)(x)

2 - (log(x)y + log(y)(x))

I went to www.quickmath.com, and i typed in 2 - ((log(y)/log(x)) + (log(x)/log(y))) = 0

This gave showed me that the problem can never be 1 or anything less than 1.

Answer 5

It's summer..take a break!