Hey guys , can you explain me why in log equation log5x = log2 + log2x +1, I can’t undo logs individually so it’s become 5x = 2 + 2x +1
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>I can’t undo logs individually so it’s become 5x = 2 + 2x +1
I’m not sure what you mean. Are you asking why you can’t just remove logs from the equation? Because you can’t. log(x) isn’t equal to x, so I’m not sure why you would be able to.
ok so mathematicians are really bad at choosing words for things. you know how addiction and subtraction are inversions of each other? like, 3 + 4 = 7 and 7 – 4 = 3 are saying the same fact from different perspectives?
well, logarithms, exponents, and radicals (you probably know radicals as “roots” as in “square root”) are saying the same fact from three different perspectives. 2 to the 3rd power is 8. log base 2 of 8 is 3. the 3rd root of 8 is 2. these are all the same fact, just written different ways.
the easiest way i’ve learned to remember it is to just revert to exponents. log base X of Y = Z is asking the question “what power Z do i have to choose so that X ^ Z = Y?”
one other note: you’ll often run into logarithms that don’t specify a base. this could be one of two things, and hopefully your teacher or the authors of your textbook will tell you which one they’re using. it could mean that the base is 10: especially in cases where you’re using scientific notation, log base 10 can be really useful. however it could also mean a very special and complicated number called e. e is around 2.718. e is basically the number which makes logarithms and exponents play well with calculus, but it has a bunch of other funky properties too.
It’s because the logarithm function is not linear. This simply means that log(a + b) ≠ log(a) + log(b). Another function lacking the property of linearity is the much more commonly known square function, where it is such a commonly made assumption that squaring is linear to the point that teachers have to explicitly tell students that (a + b)^(2) ≠ a^(2) + b^(2).
An easy way to prove that logarithms aren’t linear is by simply testing two select values. For example, we could check a = b = 1. On the left side we have log(2), and on the right side we have log(1) + log(1), which is simply 0 (x^(0) = 1 for nonzero x). Obviously, log(2) is not equal to 0, so our assumption that the logarithm was linear was demonstrably incorrect.
Simply put this is an algebra problem and algebra is hard. Calculus is easy. Algebra is where you will make the mistakes.
I’m assuming you know logarithms is the inverse of exponents. It’s now applying that fact to algebra.
What you do to one side, you do to the other.
log5x = log2 + log2x +1 => 10^(log5x) = 10^(log2 + log2x +1)
The “10^” in 10^(log2 + log2x +1) does not distribute. To “distribute” them, it needs to be 10^(log2)*10^(log2x)*10^(1). So 5x = 40x.
I usually just substitute in easy numbers to test things. It makes concepts stick better than listening to the teacher and misremembering/not remembering concepts.
10^(2+2+1) = 10000 <- original
10^2 + 10^2 + 10^1 = 100 + 100 + 10 = 210 =/= 10000 <- yours
10^2 * 10^2 * 10^1 = 100 * 100 * 10 = 10000 = 10000 <- correct
Why does it do this? I can only answer that’s how it is or that’s above my pay grade but using simple math I can show that what you did is wrong, which is enough to pass math class.
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