MHB Natural Log Rule: $\frac{a}{b}=-\frac{b}{a}$?

AI Thread Summary
The discussion confirms that the natural logarithm of a fraction can be expressed as the negative logarithm of its reciprocal, specifically stating that $\ln\left(\frac{a}{b}\right)$ equals $-\ln\left(\frac{b}{a}\right)$. Participants agree that this relationship holds true, illustrating that $\ln(a) - \ln(b)$ simplifies to $\ln\left(\frac{a}{b}\right)$ and can also be represented as $-\left[\ln(b) - \ln(a)\right]$. The conversation emphasizes that inserting a negative sign in front of the logarithm effectively flips the fraction. Overall, the properties of logarithms regarding fractions and their reciprocals are clearly established.
tmt1
Messages
230
Reaction score
0
If I have $\ln\left({a}\right) - \ln\left({b}\right)$ that would equal $\ln\left({\frac{a}{b}}\right)$ or $-(\ln\left({b}\right) - \ln\left({a}\right))$ which is also $- \ln\left({\frac{b}{a}}\right)$. So does this mean $\ln\left({\frac{a}{b}}\right)$ equals $- \ln\left({\frac{b}{a}}\right)$?
 
Mathematics news on Phys.org
Yes, you are correct...another way to think of it is:

$$\log_a\left(\frac{b}{c}\right)=\log_a\left(\left(\frac{c}{b}\right)^{-1}\right)=-\log_a\left(\frac{c}{b}\right)$$
 
tmt said:
If I have $\, \ln (a) - \ln (b)\,$ that would equal $\,\ln\left({\dfrac{a}{b}}\right)\,$ or $\,-\left[\ln\left({b}\right) - \ln\left({a}\right)\right]\;$ which is also $\,- \ln\left({\dfrac{b}{a}}\right)$
So does this mean $\,\ln\left({\dfrac{a}{b}}\right)\,$ equals $\,- \ln\left({\dfrac{b}{a}}\right)\:$?
If you discovered this while 'fooling around' with logs, good workl

Yes indeed!
If you have the log of a fraction, inserting a minus in front
will 'flip' the fraction.

That is: -\ln\left(\frac{a}{b}\right) \:=\:\ln\left(\frac{b}{a}\right)

 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Thread 'Unit Circle Double Angle Derivations'
Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Back
Top