Question

How do you use the Change of Base Formula and a calculator to evaluate the logarithm `log_9 3`?

Answer 1

With a calculator you can use `log = log_10` or `ln = log_e` with the change of base formula to find answer `0.5`

Alternatively use `log_3` to find `1/2` algebraically.

Explanation:

The change of base formula tells you that `log_a b = (log_c b)/(log_c a)`

So you can use: `log_9 3 = (log 3)/(log 9)` with common logarithms.

On a calculator this will give:

`(log 3) / (log 9) ~~ 0.4771212547 / 0.9542425094 ~~ 0.5`

Or you can use `log_9 3 = (ln 3)/(ln 9)` with natural logarithms.

`(ln 3) / (ln 9) ~~ 1.0986122887 / 2.197224577 ~~ 0.5`

Alternatively, just do the algebra using `log_3`:

`log_9 3 = (log_3 3) / (log_3 9) = (log_3 3^1) / (log_3 3^2) = 1/2`