exercise in logarithm

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exercise in logarithm

ln x/2 = (ln x)/2. How could this be solved???
Progress, not perfection...

JProgress
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Re: exercise in logarithm

y* ln( x ) = ln( x^y ) , so
1/2 * ln( x ) = ln ( x^( 1/2 ) ) = ln( sqrt( x ) )

Put that into your exercise and you get:
ln (x/2) = ln( sqrt(x) )
x/2 = sqrt(x)
x^2 / 4 = x
x*x = 4*x
x = 4

Tell me if I'm wrong, but i think this is the correct answer.
nightmair
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Re: exercise in logarithm

Yes...i think that's write.Thanks!

-- Sat Mar 29, 2014 10:44 am --

and that's a good solution too:

ln x/2 = (lnx)/2
lnx - ln2 = (lnx)/2
2ln x - 2ln 2 = 2(lnx)/2
2ln x - ln 2^2 = ln x
2ln x - ln x = ln 4
ln x =ln 4
x=4
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JProgress
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Re: exercise in logarithm

JProgress wrote:ln x/2 = (ln x)/2. How could this be solved???

x does equal 4. The above posters are correct, but they are missing a solution (which luckily does NOT work).

ln(x/2)=(lnx)/2
/* multiply both sides by 2 */
2ln(x/2)=lnx
/* manipulate left site of equation using the properties of logarithms power rule */
ln((x/2)^2)=lnx
/* simply */
ln(x^2/4)=lnx
/* raise both sides to the power of e. This gets rid of both the natural logs. */
x^2/4=x
/* multiply both sides by 4 */
x^2=4x
/* subtract 4x */
x^2-4x=0
/* pull out an x */
x(x-4)=0
thus, we get x=4 or x=0.

Let's plug back in to check our solutions... starting with x=4

ln(4/2)=ln(4)/2
/* simplify the left side of the equation and rewrite ln(4) as ln(2^2) */
ln(2)=ln(2^2)/2
/* using the power rule we can rewrite ln(2^2) as 2ln(2) */
ln(2) = 2ln(2)/2
/* simplify to right side of the equation (2 * 1/2 = 1)
we are left with ln(2)=ln(2), so x=4 is correct!

Now onto x=0
ln(0/2)=ln(0)/2
We can already stop because ln(0) DNE... When x=0 the equation DNE. This is because ln(0)=x can be rewritten as e^x=0, and no value of x satisfies that equation.
imsotired
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