Log functions are somewhat opaque at first glance, as they don’t represent a natural arithmetic process, like polynomials or even exponentials. This is because logarithms are actually constructed - at least initially - artificially as a function that is the “inverse process” of the exponential function. This is what we want to explore here.

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Exponential functions can be difficult to compute - both because numbers can get very large, but also because any values that aren’t nice integers end up being very difficult to calculate with any precision. For example, calculating something like $2^x = 8$ isn’t too hard, but it isn’t too obvious that $2^x = 131072$ means $x=17$, and much worse is solving something like $2^x = 7$, since $7$ isn’t a nice power of $2$.

This is why logarithms were invented. In particular, a logarithm with base “$b$” is defined as the inverse function to the exponential $b^x$.

The main use of logarithms then, is to undo exponential functions, just like inverse functions undo each other. In other words, one of the primary uses of logarithms is to cancel exponential functions.

Returning to our example of $2^x = 7$ we would isolate $x$ in this problem by taking a log base 2 of both sides. This gives us:

Remember that inverse functions undo each other in either order. In particular, although the above is the most common way to use logs, you can also use exponentiation to undo logs just like you can use logs to undo exponentiation. For example:

This second example may seem especially odd, as we don’t usually see logs in an exponent, and we don’t usually use exponentiate both sides of an equality. Nonetheless this is a perfectly valid technique, and importantly it’s the way one “undoes” logarithms and/or exponentials. Using logs to undo exponentials is just like dividing to undo multiplication.