Better Understanding Logarithms I am having a really hard time understanding logarithms.
My trouble comes from the fact that you can rewrite an exponential function as a logarithm, but at the same time the inverse of that exponential function is also a logarithm.
Firstly, what does it mean that these two functions are equivalent?
x = b^y
y = log_b(x)

Also, how can x = b^y have an equivalent logarithmic function (y = log_b(x)) but its inverse function is also a logarithmic one?
x = log_b(y)

 A: $x = b^y$ and $y = \log_b x$ are equivalent statements (about how $x$ and $y$ as specific variables and/or numbers) are related to each other but they are not equivalent functions.
$f: \mathbb R^+ \to \mathbb R$ via $f(x) = \log_b x$ and $g: \mathbb R \to \mathbb R^+$ via $g(x) = b^x$ are must certainly not equivalent.  They are inverses.
$x = b^y \iff y = \log_b x$ are equivalent statements in the same way $8 = 2*4 \iff 2 = \frac 84$ are equivalent statements.  The act of multiplying two numbers together is the exact opposite (inverse) of dividing two numbers.  But $8 = 2*4$ and $2 = \frac 84$ both say the same thing as "$2$ and $4$ and $8$ are related is such a way that $8$ comprises of $2$ pieces of $4$ parts".
$x = b^y$ and $y = \log_b x$ both say the equivalent statement "$x$ and $y$ are related in such a way that $b$ raised to the $y$ power results in $x$ and $y$ is the power you must raise $b$ to get the result of $x$".  They are two different ways of saying the same thing.
....
And if $x = b^y$ and $y = \log_b x$ then the third statement $x = \log_b y$ is not true and completely out of left field.
