Logby=x means
bx=y
Some examples...
Log381=4 means 34=81
Log1010000=4 means 104=10000
log41/16=-2 means 4-2=1/16
and now for some log examples in the other direction...
22=4 means log 24=2
9 1/2=3 means log93=1/2
5-3=1/125 means log51/125=-3
Things to remember about logarithms before we get going:
- Logs where the base (b) is not shown are automatically base 10.
- The log button on your calculator is automatically base 10.
- Ln automatically makes the base 'e'. (This is called the natural log.)
- WHETHER THE BASE OF YOUR LOG IS 10, e, OR SOME OTHER POSITIVE VALUE, LOGS THEMSELVES ARE JUST EXPONENTS!!!
If you find yourself struggling with this review, and even some of the concepts I will be going over in a minute, this '[logs] for dummies' site is succinct and helpful
............................................now............................................
Let's start going over how to solve logarithmic
equations!
What's great about mathematics is that there is almost always more than one way to go about solving a problem, and solving logarithmic equations is no exception to that!
Here is option #1 of how to solve logarithmic equations, (and my personal favorite, if I do say so myself...): REWRITE AS A LOG.
I will explain by using an example. Let's say we have 2x=64. My first instinct when I see this problem is to start using my second favorite method in the study of mathematics, the guess and check. But we don't have to go there, because we have the CHANGE OF BASE FORMULA.
Change of base formula: 
Because of this formula, I can solve 2x=64 by doing log64 divided by log2 to get my answer. So, overall, my work would look like this:
1. 2x=64
2. log64 / log2=6.......
3. check: 26=64
Lets try solving one more this way before we move on:
1. 42x=3111696
TAKE THE LOGS OF BOTH SIDES, AND DIVIDE
2. log3111696 / log42=4
CHECK...
3. 424=3111696
Here is option #2 of how to solve logarithmic equations: TAKE THE LOG OF BOTH SIDES.
While personally I think this method is tedious, it can be very helpful when the equations become more complicated. But we don't have to go there right now, so let's just begin with a simple example. Let's say we have 3x=729. First, take the log of both sides, so we have log(3x)=log729. Because of the power rule, loga (xp) = p loga x, we can shift our equation to be xlog3=log729. Once we have done that, the real mathematical tom foolery commences!!!!!! We continue by dividing both sides of the equation by log 3. So, once we do that we are left with x=log729 / log 3. We can use the change of base formula in a reverse kind of way to also rewrite this as x=log3729. See the work below:
1. log(3x)=log729
2. xlog3=log729
3. xlog3=log729
log 3
4. x=log729
log 3
(step four also can be written as x=log3729)
5. x=6
1. 11x=161051
TAKE THE LOG OF BOTH SIDES
2. log(11x)=log161051
USE THE POWER RULE TO BRING EXPONENTS DOWN
3. xlog11=log161051
DIVIDE BOTH SIDES BY log11
4. xlog11=log161051
log 11
SIMPLIFY
5. x=log161051
log 11
DIVIDE
6. x=5
Now, here are some websites and videos that I found to be particularly helpful.
This link basically gives some more examples of how to use the change of base formula in solving the equations.
The guy in this video talks about taking the logs of both sides....his problems are simplified a little bit more, and he is a little impatient, but I think it should help.
This link is for those of you looking for a really tricky problem.....
I hoped this post helped!
This was a very good post, i think the written explanations were what helped me the most.
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