I need some help understanding how logarithms work. I just can’t wrap my head around the concept, and I can only find videos online explaining how to rearrange a log equation. I’m looking for an explanation as to how it works, and perhaps the theory behind it.
I’ve also heard that log is easier to use for more complex calculations. How does rounding work with log? What is a natural log? What is e?
In: 3
Are you familiar with powers? You need to be comfortable with powers before ever getting to logarithms, so I’ll assume you’re good with that. Now, 2^(4) is 2 times itself 4 times, or 16. 2^(10) is 2 times itself 10 times, or 1024. Now quick, what is 16 times 1024? Well, it’s kind of a pain to multiply those two numbers, but knowing they’re powers of 2, we can simply add the exponents, right? So, 2^(4) times 2^(10) is 2^(4+10), or 2^(14). Which is obviously 16,384.
So here’s the trick with logarithms. For any given base (in our example, base 2) the logarithm of a number is the power you raise the base to to get that number. From the above, you can now see that the log (base 2) of 1024 is 10. And the log (base 2) of 16 is 4. And you can multiply 16 times 1024 by adding their logarithms.
“e” is the base of the so-called “natural” logarithms. These use Euler’s number “e” (2.718281…) as their base, because this simplifies certain operations in calculus, but the principle is the same. Don’t believe me? Let’s run the above example using base “e” instead of base 2.
log (base e) 16 = 2.7726 (Which makes sense. 2^(4) is 16, so “e”, being a slightly bigger base, takes a slightly smaller exponent to produce the same result)
Similarly, log (base e) 1024 = 6.9315
So, to multiply 16 times 1024 using natural logs, we add the exponents 2.7726 + 6.9315 = 9.7041. To find the result, we raise “e” to that power: e^(9.7041), which comes out to (surprise) 16,384.
Hope that helps.
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