SOLUTION: I need some help with two problems. Thanks alot. Solve the logrithim equation. leave answer in exact form log x = log 2x^2 -2 solve the equation ln 4 - ln x = 20 tha

Algebra ->  Algebra  -> Exponential-and-logarithmic-functions -> SOLUTION: I need some help with two problems. Thanks alot. Solve the logrithim equation. leave answer in exact form log x = log 2x^2 -2 solve the equation ln 4 - ln x = 20 tha      Log On

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Question 71654This question is from textbook
: I need some help with two problems. Thanks alot.
Solve the logrithim equation. leave answer in exact form
log x = log 2x^2 -2
solve the equation
ln 4 - ln x = 20

thanks againThis question is from textbook

Answer by bucky(2100) About Me  (Show Source):
You can put this solution on YOUR website!
log%28x%29+=+log%282x%5E2%29+-2
.
I'm going to assume that the base of log is 10 for this problem.
.
For the time being let's just work on log%282x%5E2%29. I presume you know the rules
of logarithms. log%282x%5E2%29 equals %28log%282x%29%2B+log%28x%29%29 and furthermore,
log%282x%29=+log%282%29%2Blog%28x%29
.
So putting this expansion all together:
.
log%282x%5E2%29+=+log%282%29%2Blog%28x%29%2Blog%28x%29
.
Substituting this into the original problem for the term log%282x%5E2%29 results in:
.
+log%28x%29+=+log%282%29%2Blog%28x%29%2Blog%28x%29-2
.
On the right side, combining the two log%28x%29 terms gives us:
.
log%28x%29+=+log%282%29+%2B+2log%28x%29+-+2
.
Subtract 2log%28x%29 from both sides leaves us with:
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-log%28x%29+=+log%282%29+-+2
.
Multiply both sides by -1:
.
log%28x%29+=+-log%282%29+%2B+2
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Using a calculator you can find log(2) and subtract it from 2. The answer is 1.698970004.
So the equation is now:
.
log%28x%29+=+1.698970004
.
Convert this to exponential form to find that:
.
x+=+10%5E1.698970004+=+50
.
The answer is that x = 50, but you can track the work and cut it off at the point
that you think satisfies the "exact form" asked for in the problem.
.
Next problem:
.
ln%284%29+-+ln%28x%29+=+20
.
Subtract ln%284%29 from both sides to get:
.
-ln%28x%29+=+-ln%284%29+%2B+20
.
Multiply the entire equation by -1
.
ln%28x%29+=+ln%284%29-20
.
This time the base is e.
.
Use a calculator to find ln(4) and once you get that, subtract 20. The result should be
-18.61370564. This makes the equation:
.
ln%28x%29+=+-18.61370564
.
Convert this to exponential form by raising e to the power -18.61370564 on your calculator.
The answer you should get is
.
x=+%288.24461449%29%2A%2810%5E%28-9%29%29
.
Hope this helps you with your understanding of logarithms.