I know this is a very generalized question, but to get a perspective on how they relate to one another graphically and algebraically... also what about the meaning of log10 or "e"? What are the most important techniques when solving the logarithmic and exponential equations? Thanks! Math is fun once the light goes off
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Exponentials and logarithms are inverse of each other. If you take the graph of y = 10 ^ x and flip it over the diagonal line y=x, what you get is the graph of "y = log10(x)" by which I mean the log base 10 of x.
"Log10(x)" asks the question, "To what power do I need to raise 10 to get x?" x = 10^?
For example, 100 = 10^2, so log10(100) = 2. Another example, 10000 = 10^4, so log10(10000) = 4.
Without a calculator, the log of 2734476 is going to be 6 point something (maybe just above about 6.3), because 2734476 is somewhere between 10^6 and 10^7.
You can use any base you like, and the good news is that it's trivially easy to change from one log base to another. We count in base 10, so log base 10 is very common. So common in fact that we just write log(x) when we mean log10(x). The other common base in use will be base e, which is just a magic number about equal to 2.718 in much the same way that pi is a magic number about equal to 3.14. The cool thing about log base e really shows up when you get into calculus and start doing derivatives and integrals with exponential and log functions. It's so cool, in fact, that you'll start writing ln(x) instead of log base e of x.
In computer science, we use log base 2 a whole lot. For example, I know an algorithm that will guess any integer between 0 and n in only 2 * log2(n) guesses. The maximum number has to double in size before I need another guess.
Mathematically, the whole point of logarithms is that if I have an equation like
y = 17^x
and I want to solve that for x, I can't do it with arithmetic, or even with roots like square roots, cube roots, or any base root. You need logarithms to solve for any variable in an exponent. In this case, I take the log base 17 of both sides-
"Log10(x)" asks the question, "To what power do I need to raise 10 to get x?" x = 10^?
For example, 100 = 10^2, so log10(100) = 2. Another example, 10000 = 10^4, so log10(10000) = 4.
Without a calculator, the log of 2734476 is going to be 6 point something (maybe just above about 6.3), because 2734476 is somewhere between 10^6 and 10^7.
You can use any base you like, and the good news is that it's trivially easy to change from one log base to another. We count in base 10, so log base 10 is very common. So common in fact that we just write log(x) when we mean log10(x). The other common base in use will be base e, which is just a magic number about equal to 2.718 in much the same way that pi is a magic number about equal to 3.14. The cool thing about log base e really shows up when you get into calculus and start doing derivatives and integrals with exponential and log functions. It's so cool, in fact, that you'll start writing ln(x) instead of log base e of x.
In computer science, we use log base 2 a whole lot. For example, I know an algorithm that will guess any integer between 0 and n in only 2 * log2(n) guesses. The maximum number has to double in size before I need another guess.
Mathematically, the whole point of logarithms is that if I have an equation like
y = 17^x
and I want to solve that for x, I can't do it with arithmetic, or even with roots like square roots, cube roots, or any base root. You need logarithms to solve for any variable in an exponent. In this case, I take the log base 17 of both sides-
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