Question

How do you evaluate `log_3 64`?

Answer 1

I found `3.78557`

Explanation:

I would try to change base and use a pocket calculator. The natural log, `ln`, can normally be evaluated using a pocket calculator so I 'll try to use `ln`:
`log _3(64)=(ln(64))/(ln(3))=3.78557`

Remember that to change base to a new base `c` you do:
`log_b(a)=(log_c(a))/(log_c(b))`

Answer 2

`log_3 64 = (6 log 2)/(log 3) ~~ 3.7855786`

Explanation:

Suppose you know the following approximations:

`log_10 2 ~~ 0.30103`

`log_10 3 ~~ 0.47712125`

The change of base formula tells us that:

`log_a b = (log_c b)/(log_c a)`

for any `a, b, c > 0` with `a, c != 1`

So we find:

`log_3 64 = (log_10 64)/(log_10 3) = (log_10 2^6)/(log_10 3) = (6 log_10 2)/(log_10 3) ~~ (6*0.30103)/(0.47712125) ~~ 3.7855786`