What's a log? A log is an exponent. So, when you see log_2 8, it means "what is the exponent on 2 to give you 8?" (Ans: 3)
log_2 35 =?
well, 2^5 is 32, 2^6 is 64, so log_2 35 is a number between 5 and 6, closer to 5. After that, push buttons on a calculator. (I assume you're not going to do a Taylor expansion and figure it out by hand.)
When you're multiplying x^3 times x^5, what do you do with the exponents? You add them. That's why log(xy) = log x + log y
ln means log base(Euler's number e), or log base 2.718281...
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Now, when you're solving equations, the rule is "what you do to one side, you do to the other side."
So, if you're solving something like 3x - 5 = 17, the first step most people would do is add 5 to both sides. Then, you'd divide both sides by 3. Essentially, what you're doing is chopping off bits and pieces from the side with the x until only the x remains. You do this by using inverse functions. i.e. what's the opposite of subtracting 5? Adding 5. That's why you add 5 to both sides. 3x means 3 times x. So, how do you undo multiplication? You divide. That's why you divide by 3.
The same applies for functions involving logs or exponents.
Since log means log base 10 (in most math books; context is very important for logs. In many contexts, log is accepted as meaning the natural log). Anyway, to make sure I'm clear, log base 10 of x means "what's the exponent on 10 to get x." The inverse of this is "10 to that power."
How do you get rid of a square root? You square it (both sides.)
How do you get rid of a log base 10? You "10 it", meaning on each side, you raise 10 to that power.
How do you get rid of log base e? You "e it", meaning e^(left side), e^right side.
How do you get rid of e^something? You ln both sides.
And, there's one other trick: How do you solve something like 5^x = 4872? Well, you can use one of those properties of logs someone posted above: log a^x = x log a. This follows from when you have something like (x^5)^2, you multiply the exponents.
So, for 5^x = 4872, you can take the log of both sides - either ln or log base 10. Actually, log base anything, but most calculators are limited to two choices: log (meaning log base 10) and ln (meaning log base e).
So log 5^x = log 4872. Then, bring the exponent x in front of the log and it becomes x log 5 = log 4872. Solve this the same way as you'd solve 3x = 12. The only difference is, yucky numbers. Use your calculator.
Now, one place to be careful: if you have something like log x + log y = 8, you have to be very careful about "10ing" both sides. If you did it as is, it's 10^(logx + logy). Some people think they can distribute the 10^, but you can't. It's easier to combine those two logs together first, using the product rule. So, log x + log y becomes log xy.
You have log xy = 8
"10" both sides, and it's
10^(logxy) = 10^8
the 10^ cancels out the log (just as squaring would cancel out a square root), so the left side is simply
xy = 10^8
Sorry if this isn't super clear - my wife is yelling for me to hurry up and check on the goats in the barn before we head to bed.