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A logarithm is a form of math used to help solve the following sort of problems:

#a^x = b#

The question you're asking here is to what power do I need to raise #a# to get #b#? This exact thing can be said using logarithms (as shown below):

#log_ab = x#

The relationship between logarithms and exponents is described below:

That value #a# there is what we call our base, and it can vary based on what problem you're trying to solve.

When you have a base 10, then it's convention to just drop the base from the notation, since it's implied that you're talking about a base of 10.

So #log(3)# and #log_10(3)# are one and the same thing, the same way #x# and #1x# are the same thing: they tell you the same thing, but one has superfluous information.

When you have a base #e#, you switch to #ln#, and again drop the base from your notation.

So #ln(3)# is the exact same thing as #log_e(3)# .

As you can see, #log(x)# and #ln(x)# are not the same thing! They involve the same concept, and are both logarithms, but they are still different things.

I made a video about logarithms, if you're interested:

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