First do you know what [tex]a^n[/tex] means, for n a positive integer? We can, in various ways, extend that to [tex]a^x[/tex] for x any number (and a> 0). The "logarithm", base a, is the inverse function to [tex]a^x[/tex].
That is [tex]log_a(x)= y[/tex] if and only if [tex]a^y= x[/tex]. Because our number system is "base 10", [tex]100= 10^2[/tex], [tex]1000= 10^3[/tex], etc. That is why "logarithms, base 10", also called "common logarithms" were developed.
It is fairly easy to show that, for any base, a, [tex]a^{x+ y}= (a^x)(a^y)[/tex] so that [tex]log_a(xy)= log_a(x)+ log_a(y)[/tex]. Back when I was in high school, about 20 B.C. ("Before Computers") I learned logarithms as a way of reducing multiplication to addition. For example, to multiply (3204)(8421.2), write it as [tex](3.204\times 10^3)(8.2412\times 10^3)[/tex]. The [tex](10^3)(10^3)= 10^6[/tex] is easy. To multiply (3.204)(8.4212), look up the logarithms of the two numbers in a "table of logarithms" such as
http://www.mrfteach.mb.ca/ACC/server_fi ... lesSum.pdf. (Typically such a table would be in the back of a textbook with the "sine" and "cosine" tables.)
We find that log(3.204)= 0.5057 and log(8.4212)= 0.9254 (I confess that I used the calculator that comes with Windows and then rounded to the four places typical of logarithm tables). Since log(xy)= log(x)+ log(y) I add those logarithms: 0.5057+ 0.9254= 1.4311. The "1" we combine with 6 power of 10 to get [tex]10^7[/tex]. We use the table of logarithms "in reverse" - look up 0.4311 in the body of table (or whatever number is closes) and observe that 0.4311 is the logarithm of 2.6982 (again, I "cheated" using a computer rather than doing the calculation "by hand"). So [tex](3204)(8421.2)= 2.6982\times 10^7= 269820000[/tex] approximately.
Similarly, [tex]3.784^8[/tex] is tedious to calculate "by hand", but since [tex]log(a^x)= xlog(a)[/tex] we can look up log(3.784)= 0.5780[/tex] and multiply by 8: 4.624. Now find the "anti-logarithm" of 4.624 (look up 0.624 in the body of the table to see that log(4.2072)= 0.624) which is [tex]0.624\times 10^4= 6240[/tex].
With the advent of calculators and computers, "common logarithms" are used. But "Calculus" is still useful!
And one of the things we learn in Calculus is that the derivative (rate of change) of [tex]a^x[/tex] is a constant (depends on a but not x) times [tex]a^x[/tex] again! And we can show that the constant, for a= 3, is approximately 1.10 and, for x= 2, is approximately 0.69. That tells us that there is a base, a, between 2 and 3 so that the constant is 1! We call that number "e" (and calculate that e is approximately 2.78- it is an irrational number so has and infinitely long decimal expansion) so that the derivative (rate of change) of [tex]e^x[/tex] is again [tex]e^x[/tex], the "worlds easiest derivative".
Any time we have a function, we like to have its inverse function. We define ln(x) (the "natural logarithm") as the inverse function to [tex]e^x[/tex]- [tex]y= ln(x)[/tex] if and only if [tex]x= e^y[/tex].