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It's been some time since I've been in an academic setting, but it's embarrassing for me to have to ask this all the same.

I'm trying to figure out with a %12 daily growth, over what period will it take for take for a number of units 46,000 to crest 1,000,000 units. Here's what I've come to:

  1. $$x(t) = x_0 × (1 + r) ^ t$$
  2. $$log(x(t)) = x_0t * log(1 + r)$$
  3. $$\frac {log(1000000)} {(46000 * log(1 + 0.12))}$$
  4. $$\frac{log(x(t))} {x_0 * log(1+r)} = t$$
  5. $$\frac{log(1000000)} {46000 * log(1 + 0.12)} = t$$
  6. $$\frac{log(1000000)} {46000 * log(1.12)} = t$$
  7. $$\frac{6} {46000*0.049...} = t$$
  8. $$\frac{6} {‭2,264.029...} = t$$
  9. $$t = ‭0.00265$$

Which definitely doesn't seem right. I've played with online calculators and figured out it would take somewhere between 27-28 days. What have I done wrong here?

searchengine27
  • 103

2 Answers2

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Your mistake is way up at the top, you have $\log(x(t)) = \log(x_0) + t \log(1+r)$ (the log of a product is the sum of the logs) so $t=\frac{\log(x(t))-\log(x_0)}{\log(1+r)}$.

Ian
  • 101,645
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You've gone wrong in first step.

$$\begin{align*} x(t) &= x_0 (1 + r)^t \\ \log x(t) &= \log x_0 \color{red}{+} t \log (1+r) \\ t &= \frac{\log x(t) - \log x_0}{\log (1+r)} \\ &= \frac{\log 10^6 - \log 46000}{\log 1.12} \\ &= 27.1698. \end{align*}$$

heropup
  • 135,869