I'm having a real problem understanding logarithmic equations. Especially a certain area where you have to approximate values with given logarithms. I don't even really understand what I'm talking about.
For example:
If find the approximate value of log3 8.
I've only been shown how to solve a problem like this that has an 'approximate value' that's a convenient multiple of the product of the two logarithms. Or one of the two logarithms can square itself and then conveniently go into it. I really don't get how to solve this one.
Also, I'm trying to figure out how to do this one:
Solve 10x = 152x.
log 10x = log 152x
x log 10 = 2x log 15
??
For the record. Logarithms are exponents.
Definition: The Logarithm of a number to a given base is the power to which the base must be raised to give the number.
That is, if then .
example, what is ?
why? because we have to raise the base 2 to the power 3 to get the number 8
So logarithms are the exponents we use when we want to work in a particular base. Why all the fuss? Why not just use regular exponential equations to solve problems like these.
Straight-forward-matter-of-factly answer: try solving using the techniques for regular exponential equations.
Better-more-wishful-thinking answer: Logarithms actually make life easier for the Mathematician. They can turn powers into multiples, multiples into additions, divisions into subtractions ...
The general log rules are self-explanatory of the properties of logarithms.
Rules:
Besides the definition, here are the three rules about logarithms that you MUST know.
Note: , where is Euler's constant.
Here are some rules that it's not that important for you to memorize, but they make life a whole lot easier if you do.
, where (I think we can have negative basis, but I've never worked with them, so I won't discuss them).
does not exist
----This is called "The change of base formula (for logarithms)"
... and there are a few more that i'm forgetting at the moment. Like I said, it's not imperative that you memorize these (since they can be derived from the first four laws), but it's good if you do. Particularly the change of base formula, that shows up pretty often.
EDIT: I remembered two more rules. I added them
I don't really know......normally I'd think of isolating the first x by adding the other x to the other side....but that isn't right. I don't know how you would solve for x if there are only x's and no real numbers to compare to. I've actually been trying to breeze through a couple logarithm chapters and it's no wonder I'm in the dark about how to do these problems, and what I'm doing.
well, doing logarithms is going to be hard if you don't know basic algebra, including factoring.
We can simply take the common term out, it's the reverse of expanding brackets.
Note that if you expand those brackets, you will get the original equation. From here we make the observation, that if two numbers being multiplied gives zero, it means that one number or the other is zero. So,
either or
Thus, or
How does factoring work exactly?
we search the terms to see what is common among them. In the case of it is clear that the common term is . now we simply put brackets around the expression and divide everything inside by the common term, while simultaneously multiplying the outside bracket by the common term.
since we are dividing and multiplying by the same thing, we are not changing anything, since this is essentially multiplying by 1. so now, we simplify to get
and we continue as i described above.
i did the same thing with the log equation.
we had:
it wasn't hard to see that we have 's common to both terms on the left. we have multiplying and multiplying -1
so now we pull out the common while simultaneously dividing by , we get:
the 's in the bottom of the fractions cancel with the ones at the top and we have:
now, since we have two numbers being multiplied gives zero, one number or the other is zero.
so, or
since we must have
Okay, now I just have a problem with this base e logarithm problem.
8 + 3e^3x = 26 (subtract 8)
3e^3x = 18 (divide the 3)
e^3x = 6 (divide the other 3?)
e^x = 2 (inverse property of equality)
x = ln2
x = .6931
The actual answer is .5973 and I can't seem to find out what I did wrong. Any help, please?