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