I have a log question which asks me to express the log of a single number?
The examples I have seen are in both decimal answers and fractions, I am wondering what way the answer to a log question should be given?
$\displaystyle \dfrac{\log_9(30)}{\log_9(3)} = \dfrac{\log_9(10)+\log_9(3)}{\log_9(3)} = \dfrac{\log_9(10)}{\log_9(3)} + 1$
Since $\displaystyle 3 = 9^{1/2} \text{ then } \log_9{3} = \log_9(9^{1/2}) = \dfrac{1}{2}$ we can put this into our equation above:
$\displaystyle \dfrac{\log_9(10)}{\log_9(3)} + 1 = 2\log_9(10) + 1$
edit: this is approximately equal to 3.1 but it is better to leave it in exact form to avoid rounding errors. If this were my question I'd leave it in the form above: $\displaystyle 2\log_9(10) +1$
This is ideal, however the OP should check with their teacher. I remember in one of my previous modules, anything with logs in the final answer was not deemed "fully simplified" - the mark scheme would specify that either a decimal or a fractional equivalent were acceptable. And I won't even talk about the mechanics...
I believe the methods shown have given me a more in depth understanding of the log system although I had already previously worked out the solution myself, the members of this forum that have given clarification for me I would like to say thank you for your support.
I have contacted my tutor for clarification with regards how to present the answer, so will have to wait until I get a response, which I know will be soon, so I will advise of what I am told later.