Evaluate the following?

Question

(a) log(with a small 2) 32

(b) log(with a small 3) 6 to 3 d.p.

Answer 1

(a) log (base 2) 32 = 5 since 2^5 = 32.

(b) log (base 3) 6 can be figured, but it is not quite as easy as the first problem.

Notice this, 6 = 2 x 3.

Therefore:

log (base 3) 6 = log (base 3) (2 x 3) = log (base 3) 2 + log (base 3) 3 =

log (base 3) 2 + 1. Do you see why log (base 3) 3 = 1?

We can deal with log (base 3) 6 this way:

Write 2 as 3 raised to a power x. Then 3^x = 2.

Now we can take the natural logarithm of both sides to get this:

x ln 3 = ln 2 ---> x = ln 2 / ln 3.

So, log (base 3) 2 + 1 = (ln 2 / ln 3) + 1.

Now use a calculator to find ln 2 / ln 3.

(ln 2 / ln 3) + 1 = (0.693 / 1.099) + 1 = 0.631 + 1 = 1.631 to three decimal places.

Again using a calculator, raise 3 to the power 1.631, and you should get approximately 6.

So your answer is 1.631.

Answer 2

log base 2 of 32 is 5. 2 to the 5th is 32.
I'm not sure what you mean by "to 3 d.p."

Answer 3

a. 5
b. 1.631

this can be done in excel with
=LOG(number,base)