Exponential grown Calculation

Question

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?

Answer 1

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 $t=\frac{\log(x(t))-\log(x_0)}{\log(1+r)}$.

Answer 2

You've gone wrong in first step.

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