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

In summary, the rule for logarithms states that the logarithm of the power of a number is equal to that power multiplied by the logarithm of the number. This can also be expressed as x = log_a(y), where y = a^x. The definition of logarithm may vary, but it is commonly defined as the power to which a base must be raised to yield a given number.
  • #1
Euler
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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?
 
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  • #2
[tex]\log (x^a) = a\log x[/tex]

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.
 
  • #3
If [itex]y= a^x[/itex] then [itex]x= log_a(y)[/itex]. That is often used as the definition of the logarithm. Exactly how was "[itex]log_a(x)[/itex]" defined in your class?
 
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  • #4
HallsofIvy said:
If [itex]y= a^x[/itex] then [itex]x= log_a(y)[/itex]. That is often used as the definition of the logarithm. Exactly how was "[itex]log_a(x)[/itex]" 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.
 
  • #5
Euler said:
How can x = the log of x?

Who said ##x=\log(x)##?
Euler said:
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?
 
  • #6
adjacent said:
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.
 

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