4

Logarithms are defined as 'a quantity representing the power to which a fixed number (the base) must be raised to produce a given number' And exponents are defined as 'a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression'

So is it not true that logarithms and exponents the same and logarithmic functions and exponents are inverse of each other?

  • 5
    Logarithms and exponents are inverse to each other, but how does that imply then that logarithms and exponents are the same? – NickD Nov 01 '18 at 15:05
  • More specifically, there are self-inverse functions. Those are precisely the functions with the property: $$f(f(x))=x$$ example: $f(x)=1/x$. The exponential function is not self-inverse – Yuriy S Nov 01 '18 at 15:08
  • Am I right in saying that a logarithm is 'the power to which a base is raised to equal a particular number' –  Nov 01 '18 at 15:15
  • 1
    You are correct with regards to the above comment. The term logarithm and the term power are used to represent an exponent. However, the power of 10 and log(10) are different. The term logarithm is almost always used to refer to the log function to some base. – NoChance Nov 01 '18 at 15:52

1 Answers1

3

I decided to flesh out my comment into an answer. This is a good question.

As functions, exponentiation and logarithms are inverse to each other. For instance, if $f(x) = a^x$, and $g(x) = \log_a(x)$, then $f(g(x)) = x$ and $g(f(x)) = x$. (Slight caveat: the $x$ in the first equation must be positive, while the $x$ in the second equation can be any real number).

But exponential and logarithmic equations have the same amount of information in them. That is, $$ a^b = c \text{ exactly when } b = \log_a c $$ Here $a$ and $c$ are positive numbers, while $b$ can be any real number.

In some elementary school curricula they teach fact families. For instance $3+4 = 7$ is in the same family as $4+3=7$, $7-3=4$, and $7-4=3$. If you like, you can say that $a^b =c$ and $b =\log_a c$ belong to the same fact family, like two sides to the same coin.

Matthew Leingang
  • 26,112