1. logs and e^x

Let M be a large real number. Explain briefly why there must be exactly one root A of the equation Mx = e^x with A > 1. Why is logM a reasonable approximation to A? Write
A = logM + y. Can you give an approximation to y, and hence improve on logM as an approximation to A?

Im not sure whether what I've done is correct:

The graph of y=Mx and y=e^x only intersects once, as e^x is a much greater increasing function.

M is a large number, so M=e^x approximately, so lnM = x approximately

A = logM + y

so y=logA

Thanks

2. Originally Posted by Aquafina
Let M be a large real number. Explain briefly why there must be exactly one root A of the equation Mx = e^x with A > 1. Why is logM a reasonable approximation to A? Write
A = logM + y. Can you give an approximation to y, and hence improve on logM as an approximation to A?

Im not sure whether what I've done is correct:

The graph of y=Mx and y=e^x only intersects once, as e^x is a much greater increasing function.
You actually looked at the entire graph and not just a limited part? I don't believe that! What do you know about derivatives? Can you show that the derivative of e^x is larger than the derivative of Mx? You say "e^x is a much greater increasing function" but that is only true for very large x. You should be able to say exactly for what values of x the slope of Mx is larger than the slope of e^x and for what values of x it is less.

M is a large number, so M=e^x approximately
That doesn't even make sense. M is a constant and e^x depends on x. It certainly is NOT true if x is any small number!

, so lnM = x approximately

A = logM + y

so y=logA
NO! That would be saying "A= log M+ log A" and you have no reason to think that.

Thanks

3. Originally Posted by HallsofIvy
You actually looked at the entire graph and not just a limited part? I don't believe that! What do you know about derivatives? Can you show that the derivative of e^x is larger than the derivative of Mx? You say "e^x is a much greater increasing function" but that is only true for very large x. You should be able to say exactly for what values of x the slope of Mx is larger than the slope of e^x and for what values of x it is less.

That doesn't even make sense. M is a constant and e^x depends on x. It certainly is NOT true if x is any small number!

NO! That would be saying "A= log M+ log A" and you have no reason to think that.
Ok so I have thought about this again, and now by thinking of the graph of y = Mx - e^x

It has a maximum point, and 2 roots, 1 between 0 to 1 and the other which occurs later. So the root we are interested will be the later one.

Am i right so far?

Also, can you please give me a hint on the next bit, the logM approximation.