Obviously [imath]y> 1[/imath], and since [imath]f(y) = y \log_2 y[/imath] is monotonically increasing there is only one solution to the equation.lets hope someboby will give us the proof of the problem:
if [math]ylog_2y=24[/math] then y=8
If the problem statement is (or something similar):lets hope someboby will give us the proof of the problem:
if [math]ylog_2y=24[/math] then y=8
If the problem is exactly as stated, then you need both to show that y=8 is a solution, and to show that there is no other.lets hope someboby will give us the proof of the problem:
if [imath]ylog_2y=24[/imath] then y=8
a full formal proof is required which i am not aware ofYou put this under Algebra, so it isn't clear whether you can use calculus to show that the LHS is monotonically increasing (for x greater than a certain small number); but if a full formal proof isn't required, it may be enough to write the equation as [math]log_2(y)=\frac{24}{y}[/math] and observe that the LHS is increasing while the RHS is decreasing for x>0, so there can be only one intersection of the graphs.
When we are asked to prove that 2x+1=5 we do not just put into equation x=2If the problem statement is (or something similar):
Prove that y = 8 is a solution of \(\displaystyle y * \log_2(y) \ = \ 24 \)
Then simply replace 'y' by 8 in the given equation.
Alternately, one can claim that a product of two positive increasing functions is an increasing function. See Mom: no calculus !and observe that the LHS is increasing while the RHS is decreasing for x>0, so there can be only one intersection of the graphs.
True. But also, we don't know whether @chrislav has a theorem asserting that, or has to prove it (using the definition of an increasing function).Alternately, one can claim that a product of two positive increasing functions is an increasing function. See Mom: no calculus !
Proof depends on context; what would be accepted as a "full formal proof" depends on the course you are taking.a full formal proof is required which i am not aware of
When we are asked to prove that 2x+1=5 we do not just put into equation x=2