SOLUTION: log (base 2) 40 = x

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Question 267919: log (base 2) 40 = x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%282%2C+%2840%29%29+=+x

There is no calculator I know of which can calculate base 2 logarithms. So how do we figure this out? Answer: We have to change the base to one our calculator "knows", like base 10 or base e (ln) logarithms. There is a formula for this but I always have trouble remembering it. So here's how I would do this problem. First, I'll rewrite the equation in exponential form:
40+=+2%5Ex
Now we'll find the base 10 (or base e) logarithm of each side:
log%28%2840%29%29+=+log%28%282%5Ex%29%29
Next I'll use the property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent out in front:
log%28%2840%29%29+=+x%2Alog%28%282%29%29
Now I'll divide both sides by log(2):
log%28%2840%29%29%2Flog%28%282%29%29+=+x
Note 1: This is what you would have gotten if you used the base conversion formula for logarithms.
Note 2: If you used base e logarithms you would have:
ln%2840%29%2Fln%282%29+=+x
and, believe it or not, you will get the same answer from both equations.
Note 3: log(40)/log(2) IS NOT log(20)!!

Now just use your calculator to find the two logarithms and divide. You should get something close to 5.322.