Thread: Exponential Algebra

Trying to help my brother with this tough one but neither of us can get the ball rolling, some hints at the first few steps would be much appreciated.

Two functions, y=a^x and y=logax (base a, sorry cant figure the formula codes)

Now solve for the value a where the 2 curves just touch each other and what are the coordinates of this point of intersection.

Its the first few steps were having the trouble with.
Any help would be great.

Originally Posted by manny_sr

Trying to help my brother with this tough one but neither of us can get the ball rolling, some hints at the first few steps would be much appreciated.

Two functions, y=a^x and y=logax (base a, sorry cant figure the formula codes)

Now solve for the value a where the 2 curves just touch each other and what are the coordinates of this point of intersection.

Its the first few steps were having the trouble with.
Any help would be great.

Curves intersect: $\displaystyle a^x = \log_a x$ .... (1)

Same gradient: $\displaystyle (\ln a) a^x = \frac{1}{x}$ .... (2)

You require the simultaneous solution to equations (1) and (2) (equation (2) forces the curves to touch at just a single point - why?)

I'd suggest an approximate solution is required. Try using technology to solve the above equations.