How does e^(2ln(t)) equal t^2?

  • Thread starter Thread starter CinderBlockFist
  • Start date Start date
AI Thread Summary
The discussion centers on the confusion regarding the equation e^(2ln|t|) and its equivalence to t^2. The key point is the application of logarithmic properties, specifically that a ln b = ln(b^a) and e^(ln a) = a. By correctly applying these rules, e^(2ln|t|) simplifies to e^(ln(t^2)), which equals t^2. The misunderstanding arose from incorrectly manipulating the exponential and logarithmic functions. Clarifying these properties resolved the confusion and confirmed the equality.
CinderBlockFist
Messages
86
Reaction score
0
ok guys, i don't see how e^2ln|t| = t^2 can someone explain it to me please? Seems so easy but i don't see it.
 
Physics news on Phys.org
ok, so far i tried this. I know e^(lnx) = x


so, i broke the e^(2ln|t|) into to parts:


e^2 times e^(ln|t|) which equals t (from the top identity)


so I am left with e^2 times t. which is te^2

but the book says it equals t^2..so what happened to the e? (exponential function)
 
CinderBlockFist said:
ok, so far i tried this. I know e^(lnx) = x


so, i broke the e^(2ln|t|) into to parts:


e^2 times e^(ln|t|) which equals t (from the top identity)


so I am left with e^2 times t. which is te^2

but the book says it equals t^2..so what happened to the e? (exponential function)

you have

e^2 e^(ln t) = e^(2+ln t),

which is incorrect.


you want to use the properties:


a ln b = ln b^a

and

e^ln a = a.


the rest should be straightfoward.
 
Last edited:
First look at this rule for logarithms:

\log_b(a^x)=x\log_b(a).

Now apply that to your exponent:2\ln |t|. What do you get?

Then note that f(x)=\ln (x) and g(x)=e^x are inverse functions, which means that f(g(x))=g(f(x))=x.

Those two rules together will give you the answer.
 
SWEEET! THANK YOU GUYS. I got my laws of exponents mixed up.
 
Thread 'Variable mass system : water sprayed into a moving container'
Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
Back
Top