"What is a logarithm?" my child asks. The simple answer is that a logarithm is "the opposite of an exponent". But that leads to a whole discussion of opposites.
You can think of 2 and -2 as opposites. Adding 2 then adding -2 gets you back the number you started with. That's called the negation and every number has one. 2 is the negation of -2 and -2 is the negation of 2. If you add any number to its negation you get 0. 5+ (-5) = 0. If you added 7 to something and want to undo it, you add -7 to get back where you started. So when doing addition the negation is a number's opposite.
When multiplying we have another kind of opposite. The numbers 2 and 1/2 are reciprocals. The reciprocal is the number you multiply a number by to get 1.
Both negations and reciprocals are opposites of specific numbers. Square roots and logarithms are opposite (or inverse) functions to the functions x2 and 10x.
"Isn't a square root (or cube root or whatever) the opposite of an exponent?" In one sense yes. Squaring and square rooting undo each other.
The square root of a number, n, is usually defined as "What number would you square to get n?"
so the square root of something squared is that something. √ 32 = 3.
In the language of functions, if f(x) = x2 and g(x) = √ x then f(g(x)) = ( √ x)2 = x. Since "doing" both functions gets you back to your input, the two functions are inverses. They undo each other.
Similarly log10(number) asks "what do I use as an exponent on 10, to get number?" And the functions log10(x) and 10x are inverse functions. You can have a logarithm with any "base". log7(49) would be read as "log base 7 of 49" and is equal to 2. If no base is listed, it's probably a log base 10, unless you are in a computer science class where log2 is very common. loge(x) is often writen ln(x) for "natural logarithm".