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Please help me out with this!!!

Please help me out with this!!!-example-1
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Answer:


Explanation:

Remark

I don't know how to label these so that you understand how to answer it. You are going to have to work down on each one.

ln(x + 5) = ln(x + 1) + ln(x - 1)

ln(x + 5) = ln(x^2 - 1) Two logs added means the numbers of the logs can be multiplied.

e^ln(x + 5) = e^ln(x^2 -1) e ^ln(x)= x so ...

x + 5 = x^2 - 1 Subtract x + 5 from both sides

0 = x^2 - x - 6 Factor.

0 = (x - 3)(x + 2) Solve for x

x - 3 = 0

x = 3

x + 2 = 0

x = - 2

You have to handle this extremely carefully. The minus 2 looks like it should work, but it doesn't. It is math at its best at hiding from you. Look at the last term in the given answer. ln( x - 1). If you put in - 2 for x you get ln(- 3). If you put that into your calculator, it will have all sorts of fits on you giving you every error message it can. Conclusion: you can't take the log or ln of a minus number. The answer to this is only 3 works.

e^(x^2) = e^(4x + 5)

The question means that the powers must be equal because the bases are equal.

x^2 = 4x + 5 Transfer the right side to the left.

x^2 - 4x - 5 = 0 Factor

(x - 5)(x + 1) = 0 Solve

x - 5 = 0

x = 5

x + 1 = 0

x = - 1

Again you are going to have to be a little careful. The 5 is fine. What about the - 1.

e^(-1)^2 = e^1

e^(4x + 5) = e^(-4 + 5) = e^1 And everything checks. No foul up.

Answer: - 1,5

log_4(5x^2 + 2) = log_4(x + 8)

this is the same thing as 4^m = 4^n

where m and n are thus equal because the bases are the same so the powers must be. Therefore

5x^2 + 2 = x + 8 Transfer the right side to the left.

5x^2 - x + 2 - 8 = 0 combine like terms.

5x^2 - x - 6 = 0

(5x - 6)(x + 1) = 0

5x - 6 = 0

x = 6/5

x + 1 = 0

x = - 1

The answer is (- 1 , 6/5)

log(x - 1) + log(5x) = 2

log(x - 1)*5x = 2 When you take the antilog of this you get

10^log(x - 1)(5x) = 10^2

(x - 1)*5x = 100 Divide both sides by 5

(x - 1)* x = 20 Remove the brackets.

x^2 - x = 20 Subtract 20 from both sides.

x^2 - x - 20 = 0 Factor

(x - 5)(x + 4)

x = 5 is all that works. I'm leaving you with the reason - 4 does not. Hint go back to the original question. You will again be taking the log of a negative number.

Answer: Only 5

User Rajaraman
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