SOLUTION: I have a couple of questions that I need help with: 1. Solve for x: ln7 + ln(x-3) = ln(x + 15) 2. Solve for x: 28 = 19 + 12^(x-7) **The (x-7) is an exp

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: I have a couple of questions that I need help with: 1. Solve for x: ln7 + ln(x-3) = ln(x + 15) 2. Solve for x: 28 = 19 + 12^(x-7) **The (x-7) is an exp      Log On


   

Question 4694: I have a couple of questions that I need help with:

1. Solve for x:
ln7 + ln(x-3) = ln(x + 15)

2. Solve for x:
28 = 19 + 12^(x-7)
**The (x-7) is an exponent for the number 12

3. Solve for x:
e^(3x-4) = 1/e^(x-12)
**The (3x-4) and (x-12) are both exponents

4. Use the properties of logarithms to write 3lnx^2 + ln(2x-4)-ln(3x-6) as a single logarithm.

To whoever is kind enough to help with answering these problems: Thank you and I would appreciate it if you could show your work so I can better understand how to do these types of problems myself. Thanks again!
Found 2 solutions by rapaljer, longjonsilver:
Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
1. ln7 + ln(x-3) = ln(x + 15)
First get the variables on one side (the right side) by subtracting ln(x-3) from each side.
ln 7 = ln(x+15) - ln(x-3)

Use the property of logs to express the right side as a log of a single quantity: +ln+M+-+ln+N+=+ln+%28M%2FN%29+
ln+7+=+ln+%28%28x%2B15%29%2F%28x-3%29%29+

Because ln A = ln B, you can say that A=B. In the same way,
7+=+%28x%2B15%29%2F%28x-3%29+

Multiply both sides of the equation by (x-3):
7%28x-3%29+=+%28x%2B+15%29+
7x+-+21+=+x+%2B+15+
6x+=+36
x=6

Check to make sure x= 6 doesn't give you the log of a negative. It does not.


2. 28+=+19+%2B+12%5E%28x-7%29
Subtract 19 from each side of the equation:
9+=+12%5E%28x-7%29

Take the ln of each side:
ln+%289%29+=+ln+%2812%5E%28x-7%29%29+

By law of logarithms:
ln+%289%29+=+%28x-7%29+%2A+ln+%2812%29

By distributive property:
ln+%289%29+=+x+ln%2812%29+-+7+ln%2812%29+

Add 7 ln (12) to each side:
ln+%289%29+%2B+7+ln+%2812%29+=+x+ln+%2812%29+

Divide both sides by ln (12)
+%28ln+%289%29+%2B+7+ln+%2812%29%29%2F+%28ln+%2812+%29%29+=+x+

I'm very sorry, that's all I have time for. Maybe someone else will do the others, or perhaps you can repost them.

R^2 at SCC

Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
1.ln7+%2B+ln%28x-3%29+=+ln%28x+%2B+15%29
ln%287%28x-3%29+=+ln%28x%2B15%29%29. Now just raise everything to the power e, to remove the natural logs:

7(x-3) = x+15
7x-21 = x+15
6x = 36
x = 6

2. 28+=+19+%2B+12%5E%28x-7%29

9+=+12%5E%28x-7%29. Take logs to the base 10:

log9+=+log12%5E%28x-7%29
log9+=+%28x-7%29log12
x-7+=+%28log9%29%2F%28log12%29
x-7 = 0.884228
so x = 7.884228

3. e%5E%283x-4%29+=+1%2Fe%5E%28x-12%29

e%5E%283x-4%29+=+e%5E%28-%28x-12%29%29
e%5E%283x-4%29+=+e%5E%2812-x%29. Now take natural logs of both sides, to remove the "e":

3x-4 = 12-x
4x = 16
x = 4

--> check these answers in the original questions.

jon.