Calculus - Tangent Line Question

[rakalakalili]

Homework Statement

Hello, this is a problem from the practice test for the GRE subject test.
For what value of b is the line $$y=10x$$ tangent to the curve $$y=e^{bx}$$ at some point in the xy-plane?
A) $$\frac{10}{e}$$
B)10
C)10e
D)$$e^{10}$$
E)e

Homework Equations

The Attempt at a Solution

For the line to be tangent to the curve, they must intersect at a point and their derivatives must be equal. So we get a system of equations:
$$10x=e^{bx}$$
$$10=be^{bx}$$
But I cannot think of a way to solve this system easily. Would the best method be to simply plug in the possible answers and check? This seems time consuming, and since this is a practice GRE question I am interested in the simplest way of solving it. Please let me know if I am over looking something, thank you!

 

[MathematicalPhysicist]

Hint:
e^bx= 1+bx+o(x^2)
and we have that (e^bx-1)/x=b+o(x)

So because the derivative of e^bx should be equal the slope of the tangent line, we concur?

I just picked the point zero randomly.

 

[lvoyster]

You have to solve two equations for x and for b:
10x = e^(bx) and 10 = b*e^(bx)

Multiplying the second eqn by x and comparing it to the first eqn gives

e^bx = 10x = x*b*e^(bx)

It follows that 1 = x*b, so x = 1/b. A little more work will reveal that x = e/10.