What Does The Logarithm of the Power of a Number Mean?

[Euler]

I'm reading a book and there is a section about rules to do with logarithms but one of them I don't understand, it is very wordy and I just can't get what it means.

It says "The logarithm of the power of a number is that power multiplied by the logarithm." I really don't understand what that means, can anyone who does break it down for me?

 

[DeIdeal]

$$\log (x^a) = a\log x$$

The logarithm "log" of the power "a" of a number "x" on the LHS, the power "a" multiplied by the logarithm [of that number] "log a" on the RHS.

 

[HallsofIvy]

If $y= a^x$ then $x= log_a(y)$. That is often used as the definition of the logarithm. Exactly how was "$log_a(x)$" defined in your class?

 

[Euler]

If $y= a^x$ then $x= log_a(y)$. That is often used as the definition of the logarithm. Exactly how was "$log_a(x)$" defined in your class?

How can x = the log of x? If we do 2^3 = 8 for example then log to the base 2 of 8 = 3, yes? I am still quite confused. I don't know why I can't understand this.

 

[adjacent]

How can x = the log of x?

Who said $x=\log(x)$?

If we do 2^3 = 8 for example then log to the base 2 of 8 = 3, yes? I am still quite confused. I don't know why I can't understand this.

yes. It is correct.
As HallsofIvy said before, if $y=a^x$ then $\log_a(y)=x$.
I understand it this way: To what power should x be raised to get y?

 

[Euler]

Who said $x=\log(x)$?

yes. It is correct.
As HallsofIvy said before, if $y=a^x$ then $\log_a(y)=x$.
I understand it this way: To what power should x be raised to get y?

I'm sorry, I misread HallsofIvy's post. I think I understand now.