SOLUTION: I would like help on these questions: log base 0.25 of sqrt2 - log base sqrt2 of 0.25 inverse of y=3*4^(x-1) + 2 thank you

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Question 823335: I would like help on these questions:
log base 0.25 of sqrt2 - log base sqrt2 of 0.25
inverse of y=3*4^(x-1) + 2
thank you
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%280.25%2C+%28sqrt%282%29%29%29+-+log%28sqrt%282%29%2C+%280.25%29%29
The key to the first problem is to recognize that 0.25 and sqrt(2) are both powers of 2:
0.25+=+1%2F4+=+2%5E-2
sqrt%282%29+=+2%5E%281%2F2%29
So by using the change of base formula to change these two logs into expressions of base 2 logs, we will get something that should simplify easily. The change of base formula:
log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29
Using this on both logs we get:

Since we already know what powers of 2 produce 0.25 and sqrt(2), we can replace all the logs:
%281%2F2%29%2F%28-2%29+-+%28-2%29%2F%281%2F2%29
Now we simplify:
%281%2F2%29%281%2F%28-2%29%29+-+%28%28-2%29%2F1%29%282%2F1%29
Multiplying:
%28-1%2F4%29+-+%28-4%29
which simplifies as follows:
%28-1%2F4%29+%2B+4
%28-1%2F4%29+%2B+16%2F4
15%2F4
3%263%2F4 or 3.75