solving an exponential equation - II

[chrislav]

lets hope someboby will give us the proof of the problem:
if $$ylog_2y=24$$ then y=8

 

[blamocur]

lets hope someboby will give us the proof of the problem:
if $$ylog_2y=24$$ then y=8

Obviously $y> 1$, and since $f(y) = y \log_2 y$ is monotonically increasing there is only one solution to the equation.

 

[Deleted member 4993]

lets hope someboby will give us the proof of the problem:
if $$ylog_2y=24$$ then y=8

If 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.

 

[Dr.Peterson]

lets hope someboby will give us the proof of the problem:
if $ylog_2y=24$ 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.

You 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 $$log_2(y)=\frac{24}{y}$$ 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.

Can you confirm the problem, and tell us the context, along with whatever thoughts you had? You know by now that we don't just give answers; we expect you to show some thinking.

 

[chrislav]

You 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 $$log_2(y)=\frac{24}{y}$$ 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.

a full formal proof is required which i am not aware of

 

[chrislav]

If 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.

When we are asked to prove that 2x+1=5 we do not just put into equation x=2

 

[blamocur]

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.

Alternately, one can claim that a product of two positive increasing functions is an increasing function. See Mom: no calculus !

 

[Dr.Peterson]

Alternately, one can claim that a product of two positive increasing functions is an increasing function. See Mom: no calculus !

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).

a full formal proof is required which i am not aware of

Proof depends on context; what would be accepted as a "full formal proof" depends on the course you are taking.

If you want help with this, you need to tell us what basic facts are available as a basis for a proof; what is your context?? Are you working from axioms, or from something more immediate?

Moreover, when you are asked for a proof, you don't expect just to "be aware of" the proof; you work to find a proof. What have you tried?

As I said, you should know by now how we work here. Give us something to work with. We've given you several ideas; try them out.

 

[chrislav]

its ok thanks i found the solution

 

[lookagain]

When we are asked to prove that 2x+1=5 we do not just put into equation x=2

You do not "prove" an equation.
Subhotosh Khan stated a fact, and he was referring to "prove" as in "demonstrate."
Also, he used the phrase "a solution," so here he is not claiming that y = 8 is the
only solution necessarily.