Properties of Logs: What is the Relationship Between ln(x) and ex?

[icystrike]

Homework Statement

the question is this:
showhttp://img80.imageshack.us/img80/8628/qnszb4.jpg
is 2$$\sqrt{3}$$

whereby e is natural logarithm base

Homework Equations

The Attempt at a Solution

 

[lurflurf]

What have you tried?
Remember
x^y=e^[x log(y)]

 

[Integral]

What do you know about the properties of logs?

 

[icystrike]

What do you know about the properties of logs?

i suppose everything.. i just can't show tat it is 2sqrt3

 

[Integral]

Do you have a text?

What are some of the properties of logs. They should be highlighted in boxes. RTFM

 

[icystrike]

http://www.math.unc.edu/Faculty/mccombs/web/alg/classnotes/logs/logprops.html

i think tats all of it?

 

[Integral]

look at the product rule:

Log(M*N) = logM + LogN

Now consider Log (M ²) = log (M*M) = log M + log M = 2log M

This is justification for the rule that is not shown on that page.

Log (M^N) = N log M

Now apply that to the exponents of your problem.

 

[icystrike]

look at the product rule:

Log(M*N) = logM + LogN

Now consider Log (M ²) = log (M*M) = log M + log M = 2log M

This is justification for the rule that is not shown on that page.

Log (M^N) = N log M

Now apply that to the exponents of your problem.

but it is just the exponent constant we can't apply the logarithm rules unless to the power.

 

[Integral]

Repeat in english please

 

[icystrike]

oh.. let me rephrase
the equation is just the the sums of 2 exponent constant thus we can't apply the any logarithm rules except to the power of the constant

 

[Integral]

You can apply the rules of logs to logs where ever they appear. Start by appling the rules to the logs which appear as exponents in your proplem.


$$e^{xlny} = e^{lny^x}$$

 

[icystrike]

You can apply the rules of logs to logs where ever they appear. Start by appling the rules to the logs which appear as exponents in your proplem.$$e^{xlny} = e^{lny^x}$$

yup i applied already
3e^(ln (1/sqrt3) ) + e^(ln sqrt3)

 

[HallsofIvy]

Homework Statement

the question is this:
showhttp://img80.imageshack.us/img80/8628/qnszb4.jpg
is 2$$\sqrt{3}$$

whereby e is natural logarithm base

Homework Equations

The Attempt at a Solution

You need: a ln(b)= ln(b^a) and e^ln(x)= x.

 

[Integral]

yup i applied already
3e^(ln (1/sqrt3) ) + e^(ln sqrt3)

now you are getting some where.

you do know that :

$$e^{lnx} = x$$

 

[icystrike]

You need: a ln(b)= ln(b^a) and e^ln(x)= x.

cool it works.
i have never seen this law before!
e^ln(x)= x

thank you HALLSOFIVY and INTEGRAL .
mind explaining this law e^ln(x)= x ?

 

[Integral]

On the page YOU linked read the POWER RULE of logs.

then understand that ln = log_e

 

[icystrike]

On the page YOU linked read the POWER RULE of logs.

then understand that ln = log_e

ya.. i know about it.
how about this?
e^ln(x)= x

 

[Sjorris]

The natural logarithm is the 'inverse' of the e-exponential. If e^x is f(x), and ln(x)=g(x), then you can rewrite the equation e^ln(x) to f(g(x)), but g(x)=f^-1(x) (it's inverse), so f(f^-1(x))=x. Or qualitatively, applying a function (the exponential) to it's inverse (the natural logarithm), or vice versa, returns it's argument ('input'), they 'undo' each other.

I'm sure there is a more rigorous and mathematically correct proof out there though.

 

[icystrike]

The natural logarithm is the 'inverse' of the e-exponential. If e^x is f(x), and ln(x)=g(x), then you can rewrite the equation e^ln(x) to f(g(x)), but g(x)=f^-1(x) (it's inverse), so f(f^-1(x))=x. Or qualitatively, applying a function (the exponential) to it's inverse (the natural logarithm), or vice versa, returns it's argument ('input'), they 'undo' each other.

I'm sure there is a more rigorous and mathematically correct proof out there though.

i do agree with your proving however until that inverse portion what do you really do to inverse it? i know it is somehow inverse of its law but how do you really come about to do it?

 

[icystrike]

i do agree with your proving however until that inverse portion what do you really do to inverse it? i know it is somehow inverse of its law but how do you really come about to do it?

btw you see, ln x means x=e^? and e^x means e x e x e x e x ...
^
l
x factors
i don't really see the relationship

 

[Mark44]

btw you see, ln x means x=e^? and e^x means e x e x e x e x ...
^
l
x factors
i don't really see the relationship

BTW, if y isn't an integer, e^x doesn't mean x factors of e.

Assuming y > 0,
$$x = ln(y) \iff y = e^x$$

That's the relationship between these two functions. From this relationship, you can compose them in either order:
$$ln(e^x) = x$$
$$e^{ln(y)}= y$$ if y > 0.

 

[Sjorris]

Ok, if I describe both functions and then describe the total equation, you might figure it out.
ln(x)=a is a function which, when applied to x, returns a value a which. This value a has the property that if you powered e to to that value a would return x. So if ln(x)=a is true, then e^a=x is true.

Now, in the equation you power e to a value ln(x), which we'll simplify to b for the moment, so ln(x)=b. Then the equation reads e^b=x. We just established that ln(x)=b means that if you powered e to b, it would return x. And that's exactly what you are doing, you are powering e to b, or e to ln(x), so the outcome is x.

PS. This is true for x>0, otherwise you are dealing with complex numbers.

 

[HallsofIvy]

btw you see, ln x means x=e^? and e^x means e x e x e x e x ...
^
l
x factors
i don't really see the relationship

No, that's not true. I recommend you look up the definitions of e^x and ln(x). There is no point in trying to do problems if you don't know the definitions of the very things you are working with!