There are natural logs to the base 2.7 written "ln(x)"
There are base 10 logs written "log(x)"
Let's work with base 10 logs until you understand them, and then you
can go to other types of logs such as natural logs or base 2 logs,
etc.
log(100) = 2
log(1000) = 3, just count the zeroes
log(100*1000)=log(100,000)=5, just count the zeroes.
so you can see that the log(a*b)=log(a) + log(b)
so you can see that the log((x+2)*(x-2))=log(x+2) + log(x-2)
so you can see that the log((x+2)*(x-2))=log(x+2) + log(x-2)=log(5)
so you can see that .....log((x+2)*(x-2))
=...........................=log(5)
so you can see that ..........(x+2)*(x-2)
=...........................=......5
x^2-4.=5
x^2=9
x=3
#2
Your teacher is asking you to suppose something that is true for all
values of x. In reality it is only true when x=4.5. Your teacher got
the second equation by falsely assuming that x equals 25. (It took me
a long time to figure out how they such a false equation.) What is
wrong? It is nonsense.
I can't make any sense out of what it says about the barber story
except that the barber shaves all men but only some of the men which
is a contradiction. I guess the connection is assuming that the
equation is valid for all solutions but not valid for 25. (for me it
was even harder to figure out this connection than to figure out that
your teacher assumed x=25) This says nothing about ln(x+y)=ln(x) + ln
(y).
ln(x+y)=ln(x) + ln(y)
ln(x+y)=ln(xy)
x+y=xy
xy-y=x
y*(x-1)=x
y=x/(x-1)
both x and y must be positive because you can't take a log of a
negative number, unless perhaps someone has come up with an imaginary
branch of math but I might not believe the answer. So x must be
greater than 1. What y can equal is an interesting question.
Check
If x=3, then y would equal 3/2 from the above equation.
So let's see if ln(3+1.5)= ln(3) + ln(1.5)
ln(4.5)=1.504077
ln(3)=1.098612
ln(1.5)=0.405465
1.504077 = 1.098612 + 0.405465 ok
So when y=x/(x-1), then ln(x+y)=ln(x) + ln(y)
But perhaps your teacher was trying to disprove this equation. It is
an easy mistake for a student to say ln(x+y)=ln(x) + ln(y) . This is
normally not true. It is only true when y=x/(x-1)
The correct equation which is always true is ln(x*y)=ln(x)+ln(y). I'm
sorry that your teacher emphasizes the wrong way. I vividly remember
40 years later my teachers writing words on the black board
saying "don't spell it like this." The wrong way that the teacher
teaches is what I remember.
Go up to the top of my message to understand why ln(x+y)=ln(x) + ln
(y) and visit websites which probably explain it better than I do.
#3
I agree with the answer that another gave that as x becomes a large
negative number, the square gets way larger than the others. I did
find something interesting last night. I tried to divide the equation
through by x. When you divide both sides of an inequality by a
negative number, the direction of the greater than sign changes.
Regards,
Brian
--- In Math4u@yahoogroups.com, "Jessica" <gophert@...> wrote:
>
> Can someone help me solve these three problems?
>
>
> #1 Suppose ln(x+2)+ln(x-2)=ln(5) and the question is : "Find `x'".
Is
> the answer a number (and if so what number) or some expression
> involving x?
>
> #2 Suppose ln (x-3)=ln(x)-ln(3) {for all x}. Then it would be true
> that ln22=ln25-ln3. But ln22=3.09; ln25=3.22;ln3=1.1 [YOU can use a
> calculator!!]. But, 3.22-1.1=2.12 NOT 3.09. What is wrong!. You must
> never forget the Barber who shaves all men and only those men who do
> not shave themselves. And since we know there is no such Barber,
what
> does that say about ln(x+y)=lnx+lny?
>
> #3 If x is a large negative number (e.g. –1000, -1000000000, your
> call) is it possible that x^2+15x<c, where "c" is any positive
> integer? Why or why not?
>
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