Challenge 1: Compute log1/4 (32^-18).
The 1/4 is the small number under the log.
let log1/4 32^-18 = x
(1/4)^x = 32^-18
(2^-2)^x = (2^5)^-18
2^(-2x) = 2^-90
-2x = -90
x = 45
To compute log1/4 (32^-18), we can use the logarithmic property that states log_a (b^c) = c * log_a (b).
Step 1: Rewrite the expression using the logarithmic property:
log1/4 (32^-18) = -18 * log1/4 (32)
Step 2: Simplify the expression further by converting the base 32 to a base that is compatible with the logarithm base of 1/4, which is 2.
32 = 2^5, so we have:
-18 * log1/4 (32) = -18 * log1/4 (2^5)
Step 3: Apply the logarithmic property again to simplify the expression:
-18 * log1/4 (2^5) = -18 * 5 * log1/4 (2)
Step 4: Determine the value of log1/4 (2).
Since 1/4 is equivalent to 2^(-2) (1/4 = 2^(-2)), log1/4 (2) is equal to -2.
Step 5: Substitute the value of log1/4 (2) into the expression:
-18 * 5 * log1/4 (2) = -18 * 5 * (-2)
Step 6: Perform the multiplication:
-18 * 5 * (-2) = 180
Therefore, log1/4 (32^-18) is equal to 180.
To compute log1/4 (32^-18), we can use the properties of logarithms to simplify the expression. The base of the logarithm is 1/4, which means we want to find the exponent to which we need to raise 1/4 to get 32^-18.
First, we can express 32 as a power of the base 1/4:
32 = (1/4)^(-2)
Using the property of exponents that (a^b)^c = a^(b*c), we can rewrite log1/4 (32^-18) as:
log1/4 ((1/4)^(-2))^(-18)
By applying the property of logarithms that log_a (a^b) = b, we can simplify the expression further:
-18 * log1/4 (1/4)^(-2)
Next, we can simplify the expression inside the logarithm:
(1/4)^(-2) = (4/1)^2 = 4^2 = 16
Substituting this back into the original expression:
-18 * log1/4 (16)
Finally, we can evaluate the logarithm using the property of logarithms that log_a (a^b) = b:
-18 * 2 = -36
Therefore, log1/4 (32^-18) is equal to -36.