Find the values of a such that f(ka^2) = kf(a) when f(x) = log(x).

Let's first understand what we need to find. We want to find values of "a" such that the equation f(kaa) = kf(a) holds true, where f(x) = log(x) and k is a constant.

Using the given function f(x) = log(x), we can rewrite the equation as follows:

log(kaa) = k * log(a)

Now, we use the properties of logarithms to simplify the left side of the equation. According to the properties of logarithms:

log(a * b) = log(a) + log(b) log(a^b) = b * log(a)

Using these properties, we have:

log(kaa) = log(k) + log(a^a) log(kaa) = log(k) + a * log(a)

Now set this equal to the right side of the original equation:

log(k) + a * log(a) = k * log(a)

Next, let's consider different cases for the value of k.

Case 1: If k = 0 Then our equation simplifies to:

log(k) + a * log(a) = 0 Since k = 0, log(k) is undefined. Therefore, there are no values for "a" when k = 0.

Case 2: If k is not equal to 0 If k is some positive number other than 0, then we can subtract log(a) from both sides of the equation:

log(k) + a * log(a) - log(a) = k * log(a) - log(a)

Simplifying further:

log(k) + (a - 1) * log(a) = (k - 1) * log(a)

Now, if a = 1, log(a) = log(1) = 0 and our equation becomes:

log(k) = 0

This can only be true if k = 1. It implies that for a = 1, any value of k is valid.

If a is not equal to 1, then for the equation to be valid, the coefficients of log(a) should be equal:

a - 1 = k - 1 a = k

Thus, if a is not equal to 1 and k is not equal to 0, "a" must be equal to "k" for the equation to hold true.

In summary, the values of a that satisfy the equation f(kaa) = kf(a) are:

- For k = 1, any value of "a" is valid. - For k ≠ 1 and a ≠ 1, "a" must equal "k".