I am to evaluate $6\log_8(4)$ without a calculator. The answer is provided as 4 but I cannot see how to arrive at 4.
Ignoring the 6 at the beginning of the expression, $log_8(4)$ can be written as $8^x=4$.
Without the use of a calculator I cannot simplify that part any further. I know that the root or 3rd root of 8 is not 4.
I'm not sure if the leading 6 helps or if I am to multiply the end result by 6 or some part of the expression by 6 while doing the working out part?
Completely stuck here. How can I arrive at 4? Granular, baby steps appreciated.
[EDIT]
I'd like to add that this is the textbook chapter I am working of. It's the absolute beginning of learning about logarithms. Looking at the comments and answer so far, there's reference to dividing logs which has not been covered in this book so far. Given the content of this chapter, I wonder if it's expected of me that I know how to solve this question?
Best Answer
Let's go step by step.
We start with the fact that $8=2^3$. That's equivalent to $2=8^{1/3}$ from which we immediately deduce that $$\log _{8} 2=\frac{1}{3}$$
Now, of course, we really wanted $\log_84$. But, as $4=2^2$ we have $$\log_84=\log_82^2=2\log_82=\frac 23$$ It follows that $$\boxed{6\log_84=6\times \frac 23=4}$$