How to prove this equation $\log^2_a(x) = \log_a(x^{\log_ax})$?

Question

I depicted both these functions on the Cartesian coordinate plane and they turned out to have the same graphs. My question is: How can I get the first function from the second one doing only algebraic steps?

Answer 1

Well, by the property of logarithm, $\log_a(x^{\log_a x}) = \log_a(x)\log_a(x) = \log_a^2(x)$.

Answer 2

Use that $$\log_a{x^r}=r\log_a{x}$$ for $$x>0,a>0,a\neq 1$$

Answer 3

Recall the following property of logarithms:

$$\log_a b^c = c\log_a b; \quad b > 0$$

Applying it to the right-hand side, you get

$$\log_a x\cdot\log_a x = (\log_a x)^2 = \log^2_a x$$