maths is very hard!?

Question

solve this equation:
x^(lg 5)+5^(lg x)=0
best answer will be given if you explain and solve it!!

Answer 1

First, you must identify lg 5 and lg x because x^(lg 5)=5^(lg x).I will demonstrate:
y=lg x => x=10^y
=> x^ (lg 5)=(10^y)^(lg 5)=5^y
5^(lg x)= 5^(lg 10^y)=5^y
so:x^(lg 5)=5^(lg x)
we have: x^(lg 5)+5^(lg x)=0
=> 2*(5^(lg x))=0
=> 5^(lg x)=0
condition: x must be more than 0 to make (lg x ) meaningful!
but 5^(lg x) is always more than 0
so: your equation hasn't a root .

Answer 2

The answers above look to be correct - just out of interest, though, I wonder if the value of
lim x->0 (x^(lg 5)+5^(lg x)) is defined? And is the limit 0?

Answer 3

introduce y to be: y=lg 5 , then by definition of log:
10^y=5
x^y + (10^y)^{log x} =0
x^y + 10^{ylogx} =0
x^y + 10^{log(x^y)} =0
x^y + x^y=0
so
the equation is equivalent to:
2x^{log 5} = 0
so the only possible solution is x=0, but
this is NOT a solution to: x^(lg 5)+5^(lg x)=0 because log(0) is NOT defined.

in short: this equation has no solution.!!!!

Answer 4

Andrew's right, there's no answer the way you wrote it.

If however it's supposed to be:

x^(log 5) - 5^(log x) = 0

Then x = 5 would work, obviously.

Answer 5

Undefined. Anything having to do with zero eqal to log is normally undefined. To check this, graph it on a calculator, and see where the function touches the x axis. You will find it never does, and is therefore undefined