$a^{\log_ax}=x$

If you are in Hong Kong and you need help for university mathematics courses, please visit www.all-r-math.com.

This time, lets talk about a really simple equality, but it scares many high school kids. The equality is

$$a^{\log_ax}=x.$$

Those kids will say, "I know logarithms, but this equality is too complicated."

I ask, "Tell me what is $\log_ax$?"

They answer, "It is the answer of $a$ to the power of something equals $x$."

I say, "Write it down?"

They write, "It is the answer of $a^{(\ )}=x$."

I ask, "What is the answer?"

They confuse, "$\log_ax$?"

I say, "Write the answer inside ( )."

They write, "$a^{\log_ax}=x$."

I say, "Now you know."

They reply, "Know what?"

I say, "The equality is nothing but just the definition of $\log_ax$."

They complain, "I still don't understand!"

I ask, "Sigh! Lets start again. Do you know logarithm?"

They say, "I know logarithms, but this equality is too complicated."