Rearrange formula with a Log in it: solve p_2=p_1+5log_2(d_2/d_1) for d_2

[jamiemac]

I have a great formula for predicting how long it will take to travel a known distance 2, based on knowing a known distance 1 and the known duration for travelling distance 1

The formula I have which works well is:
p₂ = p₁ + 5 log₂(d₂/d₁)

where
p₂ = unknown pace (unit is seconds per 500m)
p₁ = known pace (unit is seconds per 500m)... say 150 seconds)
d₂ = known distance (say 5000m)
d₁ = known distance (say 1000m)

running the formula gives:
p2 = 150 + 11.6 = 161.6 seconds
which I can then use to calculate that 5000m would therefore take 5000/500 * 161.6 = 1616 seconds (26 min and 56 seconds)

What I am trying to do is rearrange the formula so that I can calculate d2 if I know p1, p2 and d1

 

[blamocur]

Can you express [imath]\log_2(d_2)[/imath] through [imath]p_1,p_2,d_1[/imath] ?

 

[stapel]

The formula I have which works well is:
p₂ = p₁ + 5 log₂(d₂/d₁)...

What I am trying to do is rearrange the formula so that I can calculate d2 if I know p1, p2 and d1

The first step would be to subtract \(\displaystyle p_1\) over to the left-hand side. Then divide through by the constant.

What have you tried as your next step, using The Relationship between logs and exponentials?

Please be complete. Thank you!

Eliz.