# Convert 60 Linear steps to Exponential from (0 - 5)

Let's say I have 60 linear steps from 0 - 5. therefore 5/60 = 0.0833 (0, 0.0833, 0.2499,..4.91667,5) Thus creating a straight line.

Now, what would I need to do if I wanted to convert these 60 steps from 0 - 5 into exponential? what equation could I use to find any step from 0 - 5?

Obviously to find the step value for linear would be:

Step Value = (5/60)*Step Number

But I can't for the life of me figure out the exponential. Thank you for your help in advance!

If I understand you correctly, you're looking for an equation of the form $$f(x) = a \times r^s$$ which should go from $0$ when $s=0$ to $5$ when $s=60$.
Linear growth adds the same amount at each step. Exponential growth multiplies by the same ratio at each step. That means exponential growth can't start at $0$.
If you want to grow from $1$ to $5$ in $60$ steps, then you want $$5 = 1 \times r^{60}$$ so $$r = 5 ^ {1/60} \approx 1.027$$ which is about $3\%$ growth per step.
If you start at $0.1$ instead of $1$ the ratio is $(50/0.1)^{1/60} \approx 1.067$, or about $7\%$ per step.
If you start at $a$ then you want $$\frac{5}{a} = r^{60}$$ so $$r = \left(\frac{5}{a} \right)^{1/60} \approx 1.027 a ^{-1/60} .$$
• See my edit. You can start as close to $0$ as you like, and adjust the ratio at each step accordingly. I've no idea what's appropriate for a circuit. You might consider plotting the value as a function of step size to see what makes sense. Jan 6 '18 at 14:04