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Log Confusion...Please explain this equation

lg(N^2+1)===??????????????? simplify


lg(N^2+1)/lgN    ===?????????


lg x/lg y===lg x - lg y              (right or wrong ??????????)





Please simplify this equation 


I am not sure what you are asking.  Are these three different problems?


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3 Answers

Q:   lg(N^2+1)===??????????????? simplify

A:      (2+1)lgN



Q:  lg(N^2+1)/lgN ===?????????

A:    3lgN/lgN



Q:  lg x/lg y===lg x - lg y (right or wrong ??????????)

A:    wrong.    lg x - lg y = lg(x/y) per rules. OR you can derive this rule yourself.

                                                                 lg x/y = lg (x * y^-1)

                                                                          = lg x + (-1) lg y

                                                                          = lg x - lg y.


When in doubt, ALWAYS go back to the FIRST rule you learned.

If you typed the first one correctly, it does not simplify. The rules we have for logs:

log(x*y)= log(x) + log(y).

log(x/y)=log(x) - log(y).

log(x^y)= y*log(x).

log base b of x = log(x)/log(b)

The first one, as you can see, cannot simplify. We cannot use the rule with exponents, because the 1 is in the parentheses. We can't factor the inside, because N^2 + 1 does not factor. We are stuck.

The second one resembles our last rule. This is the same as Log base N of (N^2+1). Again, we are stuck because this does not factor.  

The last one is wrong. You can confirm this by plugging in any two numbers into a calculator. 

For example, log(100,000)/log(10)= 5. Log(100,000)-log(10)= 4. The rule is wrong. 

It looks like you may have copied down the first couple of problems wrong though, so you may want to check that.  

lg x/lg y===lg x - lg y (right or wrong ??????????)

This is wrong, but you have the right idea. The log property states that:

(lg x/y) = lg(x)-lg(y)


(lg x* y)= lg(x)+lg(y)


Your assumptions are incorrect: log(x * y) = log (x) + log(y). The reverse is not true. In the same way, log (x/y) = log(x) - log(y). Again, the way you have it is not the same thing, and not correct.

You can check this by plugging in numbers.

If x is 1000, and y is 100, then according to you, log(1000)/log(100) would equal log(900). log(1000) = 3, log(100) = 2, and 3/2 = 1.5. Log(900) is  NOT 1.5 (it is about 2.95).

HOWEVER log(1000/100) is equal to log(1000)-log(100). Log(10) = 1. Log(1000)-log(100)= 3-2=1.

You can confirm the rule for multiplication in the same way!

Since lg(1) = 0, by your logic, the answer to the first question would always be 0 (since anything times zero is zero). This is not true.

You're right, I did get it mixed up, sorry about the confusion John. Thanks for pointing that out Yitzhak! Jumped the gun on that one, I don't see any simple ways to simplify the given equation unless we can make an approximation as shown below.

We can use the base changing formula log10(a)/log10(b)= logb(a)

For our given problem we are given:   log(N2+1)/log(N) = logN(N2+1)

If we have large N we can make an approximation: logN(N2+1)˜logN(N2) = 2