Prove.
3/(log_2 (a)) - 2/(log_4 (a)) = 1/(log_(1/2)(a))
To prove the given equation:
3/(log_2 (a)) - 2/(log_4 (a)) = 1/(log_(1/2)(a))
We need to simplify both sides of the equation and show that they are equal.
Let's start with the left side of the equation:
3/(log_2 (a)) - 2/(log_4 (a))
To simplify, we can express the logarithms in terms of a common base, such as base 2:
3/(log_2 (a)) - 2/(log_2 (a^2))
Using the property that log_b(x^n) = n*log_b(x), we can simplify the expression further:
3/(log_2 (a)) - 2/(2 log_2 (a))
Now, we can combine the fractions by finding a common denominator:
(3 - 2)/(log_2 (a))
Simplifying further, we get:
1/(log_2 (a))
Now, let's simplify the right side of the equation:
1/(log_(1/2)(a))
Since log_b(1/x) = -log_b(x), we can rewrite the logarithm as:
1/(-log_2 (a))
Simplifying further, we get:
-1/(log_2 (a))
Therefore, the right side of the equation simplifies to:
-1/(log_2 (a))
Comparing the simplified left side (-1/(log_2 (a))) with the simplified right side (-1/(log_2 (a))), we can see that they are equal.
Hence, we have proven that:
3/(log_2 (a)) - 2/(log_4 (a)) = 1/(log_(1/2)(a))