School and Work: logarithm

Open Question: How can I change a logarithm of base 10 to base 5?

First note the change of base formula (let plain "log mean log base 10):

log512 = log 12/log 5

Also note 12 = 4•3 = 2²•3, so log 12 = log 2²•3 = 2•log 2 + log 3 = 2a + b

Further note that 5 = 10/2, so log 5 = log (10/2) = log 10 – log 2 = 1 – log 2 = 1 – a

Thus substituting from the above, log512 = log 12/log 5 = (2a + b)/(1–a)

v2 = 2^(½), and ½ = 2^(-1), so by rules of logs,

- ln 1/v2 = · · · · · · · power rule
ln (1/v2)^(-1) =
ln v2 =
ln 2^(½) = · · · · · · · power rule again
½ ln 2