How we solve this equation?

[mumairfarooq1]

How we solve this equation
x*8^x =340

 

[stapel]

How we solve this equation
x*8^x =340

You'd have to solve numerically or "with technology"; it cannot be solved algebraically (nor by whole-number guess-n-check, inspired or otherwise).

In what mathematics course did you encounter this exercise?

 

[HallsofIvy]

How we solve this equation
x*8^x =340

\(\displaystyle x(8^x)\) is a continuous function so if we can find one value of x where \(\displaystyle x(8^x)\) is less than 340, there must be an x value between them where \(\displaystyle x(8^x)= 340\). The simplest thing to do is to check half way between and see if, for that new value of x, \(\displaystyle x(8^x)\) is less than or larger than 340.

It is, for example, fairly easy to see that if x= 1, \(\displaystyle 1(8^1)= 8< 340\), if x= 2, \(\displaystyle 2(8^2)= 128< 340\), if x= 3, \(\displaystyle 3(8^3)= 1536> 340\) so there must be a root between 2 and 3. If x= 2.5, \(\displaystyle 2.5(8^{2.5})= 452> 340\) so there must be a root between 2 and 2.5. If x= 2.25, \(\displaystyle 2.25(8^{2.25})= 242....< 340\) so there must be a root between 2.25 and 2.5. Half way between 2.25 and 2.5 is (2.25+ 2.5)/2= 4.75/2= 2.375 so try that next. Keep going until you have whatever accuracy you want.

 

[mumairfarooq1]

\(\displaystyle x(8^x)\) is a continuous function so if we can find one value of x where \(\displaystyle x(8^x)\) is less than 340, there must be an x value between them where \(\displaystyle x(8^x)= 340\). The simplest thing to do is to check half way between and see if, for that new value of x, \(\displaystyle x(8^x)\) is less than or larger than 340.

It is, for example, fairly easy to see that if x= 1, \(\displaystyle 1(8^1)= 8< 340\), if x= 2, \(\displaystyle 2(8^2)= 128< 340\), if x= 3, \(\displaystyle 3(8^3)= 1536> 340\) so there must be a root between 2 and 3. If x= 2.5, \(\displaystyle 2.5(8^{2.5})= 452> 340\) so there must be a root between 2 and 2.5. If x= 2.25, \(\displaystyle 2.25(8^{2.25})= 242....< 340\) so there must be a root between 2.25 and 2.5. Half way between 2.25 and 2.5 is (2.25+ 2.5)/2= 4.75/2= 2.375 so try that next. Keep going until you have whatever accuracy you want.

Great answer, thank you