1. ## Help with logarithms

So, you probably know that the formula for exponential growth is $y=a(1+r)^t$, with $a$ being the original amount, $r$ being the percent of growth, and $t$ being the amount of time. I would like to learn to get $t$ using logarithms.

Hopefully you may be able to help. Thanks in advance!

Gobblewobble123

2. Start by taking the log of both sides, any base will do.

3. Maybe you could do a step by step example of how to solve for $x$ using logs? I am trying to teach myself and I love examples! I'm not sure exactly what to do here, so anything helps.

If I take the logs of both sides, would it look like this: $log_10 {a(1+r)}=log_10 {y}$?

Gobblewobble123

4. Maybe you could do a step by step example on how to solve the problem? Examples are awesome!

If I took the log of both sides, would it look like this: log_10 {a(1+r)}=log_10 {y}?

5. Maybe you could do a step by step example on how to solve the problem? Examples are awesome!

If I took the log of both sides, would it look like this: \log_10 {a(1+r)}=\log_10 {y}?

6. Originally Posted by gobblewobble123
So, you probably know that the formula for exponential growth is $y=a(1+r)^t$, with $a$ being the original amount, $r$ being the percent of growth, and $t$ being the amount of time. I would like to learn to get $t$ using logarithms.

Hopefully you may be able to help. Thanks in advance!

Gobblewobble123
Since any base will do, I'll pick "e".
ln(y) = ln[a(1 + r)^t] = ln(a) + t*ln(1 + r)
ln(y) - ln(a) = t*ln(1 + r)
ln(y/a) = t*ln(1 + r)
ln(y/a)/ln(1 + r) = t

7. Originally Posted by TheChaz
Since any base will do, I'll pick "e".
ln(y) = ln[a(1 + r)^t] = ln(a) + t*ln(1 + r)
ln(y) - ln(a) = t*ln(1 + r)
ln(y/a) = t*ln(1 + r)
ln(y/a)/ln(1 + r) = t
It's easier if you divide both sides by a first, then take the logarithms, but it doesn't really make any difference.

8. Originally Posted by Prove It
It's easier if you divide both sides by a first, then take the logarithms, but it doesn't really make any difference.
Funny - after having demonstrated this a few times to students today, I was so distracted by the notational difference (we use A = P(1 + r/n)^{nt}) that I skipped the first step!

While the result is the same, I like to approach problems with invertible operations by composing their inverses (in reverse order), so dividing by a should have been first