What is $a^{(\log_ab)^2}$?

Question

Can the following expression be further simplified: $$a^{(\log_ab)^2}?$$

I know for example that $$a^{\log_ab^2}=b^2.$$

Answer 1

$$a^{(\log_ab)^2}=a^{\log_a(b)\times \log_a(b)}=b^{\log_a(b)}$$

Answer 2

Use:

• $$\log_x(y)=\frac{\ln(y)}{\ln(x)}$$
• $$\exp\left[\ln\left(x\right)\right]=e^{\ln(x)}=x$$

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So, we get:

$$a^{\left(\log_a(b)\right)^2}=a^{\left(\frac{\ln(b)}{\ln(a)}\right)^2}=a^{\frac{\ln^2(b)}{\ln^2(a)}}=\exp\left[\frac{\ln^2(b)}{\ln(a)}\right]$$

Answer 3

$$a^{(\log_ab)^2}=a^{\log_a b\cdot \log_a b}=(a^{\log_a b})^{\log_a b}=b^{\log_a b}=b^{\frac{1}{\log_b a}}.$$