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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)
Matthew Morris
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  • "Also, how can x = b^y have an equivalent logarithmic function but its inverse function is also a logarithmic one?" Not sure what this means. – Qudit Nov 02 '17 at 05:56
  • $x= \log_b(y)$ is not the inverse of $y = \log_b(x)$. $f(x) = b^x$ does not have an equivalent logarithmic function. $f(x) = \log_b x$ is not equivalent (note the "x" is just a place hold and not a constant or specific value). Instead $f(x) = b^x$ has a logarithmic inverse function: $f^{-1}(x) = \log_b x$. – fleablood Nov 02 '17 at 06:27

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$x = b^y$ 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.

fleablood
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