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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!

edm
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konobyBYnight
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1 Answers1

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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 $$ r = \left(\frac{5}{a} \right)^{1/60} \approx 1.027 a ^{-1/60} . $$

Ethan Bolker
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  • Thank you! The question relates to an exponential converter in an electrical engineering circuit. The input is linear from 0 - 5 volts. The output needs to be divided into 60 steps from that range. I understand that there cannot be a value of 0 exponentially. But I need to have a value close to zero. Is it possible mathematically ? – konobyBYnight Jan 06 '18 at 12:30
  • 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. – Ethan Bolker Jan 06 '18 at 14:04