Inverse numbers: constant difference values in common logs, natural logs

[canselmo]

Hi,

I'm studying for the CFA's Investment Management Certificate and have just come across something that I cannot figure out. I have tried searching everywhere for an explanation and cannot for the life of me figure it out (my math skills are very basic BTW). In the study book the following statement is made:

...the difference from year to year in the common logs is constant at about 0.11 while it is about 0.26 for the natural logs. The inverse of these numbers is 1.28825 and 1.29693 respectively, which can both be interpreted as growth rates of just under 30%, subject to rounding error.


Everywhere I've searched for inverse numbers / reciprocal numbers gives me completely different results: 1/0.11 = 9.090909... and so on. I cannot figure out how this result can be 1.28825

Any help clarifying would be very much appreciated.

Thanks

 

[tkhunny]

Hi,

I'm studying for the CFA's Investment Management Certificate and have just come across something that I cannot figure out. I have tried searching everywhere for an explanation and cannot for the life of me figure it out (my math skills are very basic BTW). In the study book the following statement is made:

...the difference from year to year in the common logs is constant at about 0.11 while it is about 0.26 for the natural logs. The inverse of these numbers is 1.28825 and 1.29693 respectively, which can both be interpreted as growth rates of just under 30%, subject to rounding error.


Everywhere I've searched for inverse numbers / reciprocal numbers gives me completely different results: 1/0.11 = 9.090909... and so on. I cannot figure out how this result can be 1.28825

Any help clarifying would be very much appreciated.

Thanks

I agree the language is a bit clunky.

Common Logs = Logs Base 10 ==> \(\displaystyle 10^{0.11} = 1.28825...\)

Natural Logs = Logs Base e ==> \(\displaystyle e^{0.26} = 1.29693...\)

The question in my mind is were did 0.11 and 0.26 come from? Interestingly, log(1.29693) = 0.1129 and ln(1.28825) = 0.2533, so maybe they are related, somehow, but I'm not seeing it off-hand.

 

[canselmo]

I agree the language is a bit clunky.

Common Logs = Logs Base 10 ==> \(\displaystyle 10^{0.11} = 1.28825...\)

Natural Logs = Logs Base e ==> \(\displaystyle e^{0.26} = 1.29693...\)

The question in my mind is were did 0.11 and 0.26 come from? Interestingly, log(1.29693) = 0.1129 and ln(1.28825) = 0.2533, so maybe they are related, somehow, but I'm not seeing it off-hand.

Great, thanks for clarifying this. Those numbers came from a table with sales numbers from year 1 to 6 and the constant increase year on year is 0.11 log10 and 0.26 LNe.

So year 1 sales were 10,000, Log10 4, LNe 9.21, year 2 13,000, Log10 4.11, LNe 9.47, etc...