Proving the constant rate of doubling for an exponential function

[SticksandStones]

Homework Statement

Given that Q=Pa^t and Q doubles between t and t+d, prove that d is the same for all t.

Homework Equations

Q=pa^t

The Attempt at a Solution

This is what I've tried so far:

$$Q_0=Pa^t$$ and $$Q_1=Pa^{t+d}$$
Then:
$$\frac{a^{t+d}}{a^t}} \equiv 2$$

This is where I begin drawing blanks again. I want to say take the log, but I'm not sure if that is right.

If so, doesn't this give me:
$$\frac{t+d}{t} \equiv log(2)$$ ?

Then from there: $$d \equiv log(2^t)-t$$

Would that be correct?

 

[NateTG]

Hint:
$$Q^{t+d}=Q^t+Q^d$$

Alternatively:
$$\log \frac{a}{b}= \log a - \log b$$