Solve: 4^3-x = 8^x   (can you explain how to do it using natural log if able to use it in this equation) thank you

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rakesh05 eNotes educator| Certified Educator

Given problem is    `4^(3-x)=8^x`

Now taking   log of both sides we get

      `ln(4^(3-x))=ln(8^x)` .       Because `ln(m^n)=n lnm`

So,  `(3-x)ln4=xln8`

or,    `(3-x)ln(2^2)=xln(2^3)`

or,    `2(3-x)ln2=3xln2`

or,     `2(3-x)ln2-3xln2=0`

or,     `{2(3-x)-3x}ln2=0`

As  `ln2!=0` . So,    {2(3-x)-3x}=0

or,                         `2.3-2x-3x=0`     

or,                           `6-5x=0`

or,                               `5x=6`

or,                                  `x=6/5` .  Answer.

violy eNotes educator| Certified Educator

We can write the 4 and 8 as an exponential expression with base 2. 

So, we will have:

`2^(2(3-x)) = 2^(3x)`

`2^(6 - 2x) = 2^(3x)`

Since the bases are the same, we can just equate the exponents. 

`6 - 2x = 3x`

Add both sides by 2x. 

`6 = 5x`

Divide both sides by 5. 

Therefore, x = 6/5.

oldnick | Student

`4^3 -x=8^x`     we study the function:

`f(x)= 8^x +x - 4^3`     the first derivative is  `f'(x)= 8^(x)lnx +1`  the second: `f''(x)= 8^(x)ln^2x +8^(x)/x`

Now if `f'(x_0)=0`  then:   `8^(x_0)= -1/ln(x_0)`   (1)

subistutes in the  second derivative:

`f''(x_0)= -ln(x_0) -1/(x_0 ln(x_0))` `=-(x_0 ln^2(x_0)+1)/(ln(x_0))`   (2)

since `x_0>0`  the  ratio is positive, if  `0< x_0 <=1` (minimum point)

and negative if `x_0 >1` (Maximus point)

Now `f(x_0)= 8^(x_0) +x_0 -4^3` `=(8^(x_0)ln(x_0)+x_0 ln(x_0))/(ln(x_0))`  `-4^3=`

`=(-1+ln(x_0)(x_0 -4^3))/(ln(x_0))`   (3)

This last realtion  show that if has a minimum is less than zero.  

Instead if maximus to have solution  needs `x_0 >= 4^3`  but it's clear that this one doesn't accomplish relation  (1)  so if exists a point `x_0`  is to be  a minimum point.

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