Please help simplify this expression:
2 log (base 4) ^9 - log (base 2) ^3
The answer should be
log(base 2) ^3
The answer should actually be
Log (base 4) ^27
Note that since 4 = 2^2, log_2(x) = 2log_4(x)
That is, 4^n = 2^2n
So, we have
2log_4(9) - log_2(3)
= log_2(9) - log_2(3)
= log_2(9/3)
= log_2(3)
Not sure where you got log_4(27). If that's the solution given, it is bogus.
To simplify the given expression, let's use the logarithmic property of subtraction:
log (a) - log (b) = log (a / b)
Applying this property to the given expression, we get:
2 log (4) ^9 - log (2) ^3
Using the property mentioned above, we can rewrite it as:
log (4) ^9^2 / log (2) ^3
Simplifying further, we get:
log (4) ^18 / log (2) ^3
Now, let's express the logarithms with the same base:
log (2) ^2 = 1
Using this property, we can simplify the expression as:
log (2) ^18 / log (2) ^3
Since the bases are the same, when dividing logarithms with the same base, we subtract the exponents:
log (2) ^18 - 3
Simplifying the exponent, we get:
log (2) ^15
Therefore, the simplified expression is:
log (2) ^15