How do you evaluate #log_(4) 66 #?

1 Answer
EZ as pi
Jun 12, 2016

#x = log66/log4#

Using a calculator gives #x = 3.022#

Explanation:

The answer is not going to be an integer. The question is asking:
"To what power must 4 be raised, to equal 66?"

The powers of 4 are : 4, 16, 64, 256 ....
66 is not one of them, but we can see that the required index will be 3. .....
log from is interchangeable with index form:

If #log_4 66= x " then " 4^x = 66#

Let's use the index form and find the log of both sides, because the variable is in the index.
#log4^x = log 66#

#xlog 4 = log66#

#x = log66/log4#

Using a calculator gives #x = 3.022#

This same result could be found from the change of base law:

#log_a b = log_10 b/log_10 a#

Some calculators can work with any base, but for those that only use base 10, this works fine.1

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