does log base of 4 sqrt(t/s)=1/2log base of 4 t - 1/2log base of 4s?
sqrt=square root
thanks
yes
yah thanks!
To determine if the given equation log base 4 of sqrt(t/s) = 1/2 log base 4 t - 1/2 log base 4 s is true, we need to use logarithmic properties and algebraic manipulation.
First, let's rewrite the equation using the properties of logarithms. The rule states that log base a of b = x can be rewritten as a^x = b.
Rewriting the equation:
4^(1/2 log base 4 t - 1/2 log base 4 s) = sqrt(t/s)
Now, let's simplify the equation further using exponent properties. We know that (a^b)^c = a^(b*c), so applying this property:
4^(1/2 log base 4 t - 1/2 log base 4 s) = 4^((1/2 log base 4 t) - (1/2 log base 4 s))
From the property of logarithms, log base a of b - log base a of c = log base a of (b/c). Applying this property:
4^(1/2 log base 4 t - 1/2 log base 4 s) = 4^(log base 4 (t/s)^(1/2))
Using the property a^(b*c) = (a^b)^c, we have:
4^(1/2 log base 4 t - 1/2 log base 4 s) = (t/s)^(1/2)
Now, let's simplify both sides of the equation. Recall that (a^b)^(1/c) = a^(b/c):
4^(1/2 log base 4 t - 1/2 log base 4 s) = (t/s)^(1/2)
4^(1/2 log base 4 t - 1/2 log base 4 s) = (t^(1/2))/(s^(1/2))
Now, let's rewrite the left side of the equation as a single logarithmic term using the properties of logarithms. Recall that a^(log base a of b) = b. Applying this property:
(t/s)^(1/2) = (t^(1/2))/(s^(1/2))
Thus, we have:
(t/s)^(1/2) = (t^(1/2))/(s^(1/2))
Since both sides of the equation are equal, we can conclude that the original equation log base 4 of sqrt(t/s) = 1/2 log base 4 t - 1/2 log base 4 s is true.
Therefore, the answer is yes, the equation is true.