Please simplify the expression:
2log(base 4)9 - log(base 2)3
very detailed solution:
let 2log4 9 = x
log481 = x
4^x = 81
2^(2x) = 81
let log2 3 = y
2^y = 3
2^4y = 3^4 = 81
then 2^(2x) = 2^(4y)
---> 2x = 4y
x = 2y
So 2log4 9 - log2 3
= x - y
= 2y-y
= y
= log2 3
log base 4 of 9 + log base 4 of 6
To simplify the expression 2log(base 4)9 - log(base 2)3, we can use the logarithmic property that states:
log(base a)b - log(base a)c = log(base a)(b/c).
Applying this property, we can simplify the expression as follows:
2log(base 4)9 - log(base 2)3
= log(base 4)9^2 - log(base 2)3
= log(base 4)81 - log(base 2)3.
Next, we can further simplify this expression by converting both logs to the same base. Let's convert them to the base 4 log:
log(base 4)81 = log(base 4)3^4 = 4.
Now the equation becomes:
4 - log(base 2)3.
Therefore, the simplified expression is 4 - log(base 2)3.
To simplify the expression 2log(base 4)9 - log(base 2)3, we can use logarithmic properties.
First, let's simplify log(base 4)9. We can rewrite this as log9/log4 using the logarithmic property log(base a)b = logb/loga. Using the change of base formula, we have log9/log4 = log(9)/log(4).
Next, let's simplify log(base 2)3. We can rewrite this as log3/log2 using the logarithmic property log(base a)b = logb/loga. Again using the change of base formula, we have log3/log2 = log(3)/log(2).
Now, let's substitute these simplified expressions back into the original expression: 2log(base 4)9 - log(base 2)3 becomes 2(log(9)/log(4)) - (log(3)/log(2)).
To simplify further, we can multiply the numerator and denominator of the first term by log(4), and also multiply the numerator and denominator of the second term by log(2). This gives us: (2log(9)log2)/(log4log2) - (log3log4)/(log2log4).
The denominator log4log2 and log2log4 cancel each other out, leaving us with: 2log(9)log2 - log3log4.
And that is the simplified form of the expression: 2log(9)log2 - log3log4.