evaluate:
log base 1/4 (1/16)^4
Log1/4 (1/16)^4 = x.
4*Log1/4 (1/16) = x
Divide both sides by 4:
Log1/4 (1/16) = x/4
Exponential form:
(1/4)^(x/4) = Log(1/16)
Take Log of both sies:
(x/4)*Log(1/4) = Log(1/16)
Divide both sides by Log(1/4):
x/4 = Log(1/16) / Log(1/4) = 2
X = 8.
Sure, let's evaluate that expression. But we'd better hold on tight, because we're about to take a journey into the world of logarithms!
Now, the expression log base 1/4 of (1/16)^4 can be rewritten as log base 1/4 of 1/256.
To find the value of this expression, we need to ask ourselves, what power do we need to raise 1/4 to in order to get 1/256?
Well, 1/4^2 = 1/16, and 1/4^4 = 1/256. So, log base 1/4 of 1/256 is equal to 4.
In other words, the value of log base 1/4 of (1/16)^4 is 4. Mission accomplished!
To evaluate the expression log base 1/4 (1/16)^4, we'll break it down step by step.
Step 1: Simplify the exponent.
(1/16)^4 = (1^4) / (16^4) = 1 / 65536.
Step 2: Rewrite the original expression as a logarithmic equation.
log base 1/4 (1/16)^4 = x.
Step 3: Convert the logarithmic equation into an exponential equation.
1/4^x = 1/65536.
Step 4: Solve for x.
To make the base the same on both sides, rewrite 1/4 as (1/2)^2.
(1/2)^2^x = 1/65536.
We can convert both sides to a common base, such as 2.
2^(-2x) = 1/65536.
To simplify the right side, we can rewrite 65536 as 2^16.
2^(-2x) = 1/2^16.
Now, we can equate the exponents.
-2x = -16.
To solve for x, divide both sides by -2.
x = (-16) / (-2) = 8.
So, the value of log base 1/4 (1/16)^4 is 8.
To evaluate the expression log base 1/4 (1/16)^4, we need to simplify the expression inside the logarithm first.
(1/16)^4 can be simplified by raising 1/16 to the power of 4.
(1/16)^4 = (1^4)/(16^4) = 1/256
Now we have log base 1/4 of 1/256.
To evaluate this logarithm, we need to rewrite it in exponential form. The logarithmic equation log base a (b) = c is equivalent to the exponential equation a^c = b.
In this case, log base 1/4 of 1/256 is equivalent to (1/4)^c = 1/256.
To find the value of c, we need to solve the equation. We know that 1/4 raised to any power will never equal 1/256 because 1/4 represents a larger number than 1/256.
Therefore, the equation has no solution, and we cannot evaluate log base 1/4 (1/16)^4.