No. of real solution of exponential equation

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In summary, there is only one real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$ This can be shown by comparing the functions $f(x)=1+8^x+27^x$ and $g(x)=2^x+9^x+12^x$ and analyzing their behavior for different values of $x$. It is evident that $f(x)$ is always greater than $g(x)$ for positive values of $x$, equal to $g(x)$ for $x=0$, and tends towards 1 as $x$ approaches negative infinity. Since
  • #1
juantheron
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No. of real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$
 
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  • #2
jacks said:
No. of real solution of the equation $1+8^x+27^x = 2^x+12^x+9^x.$
$1+8^x+27^x =1+8^x+ (2+12+9+4)^x----(1)$
$=1+8^x+(2^x+12^x+9^x)+4^x+-----$
$=2^x+12^x+9^x----(2)$
if $x\neq 0$ then (1) > (2)
$\therefore x=0$ is the only solution
 
  • #3
Albert said:
$1+8^x+27^x =1+8^x+ (2+12+9+4)^x----(1)$
$=1+8^x+(2^x+12^x+9^x)+4^x+-----$
$=2^x+12^x+9^x----(2)$
if $x\neq 0$ then (1) > (2)
$\therefore x=0$ is the only solution

You have employed a mistake known as "The Freshman's Dream"...:D
 
  • #4
MarkFL said:
You have employed a mistake known as "The Freshman's Dream"...:D
I did not say:$27^x=(2+9+12+4)^x=2^x+9^x+12^x+4^x$
I said :$27^x=2^x+9^x+12^x+4^x+----$
the remaining terms are omitted
 
  • #5
Albert said:
I did not say:$27^x=(2+9+12+4)^x=2^x+9^x+12^x+4^x$
I said :$27^x=2^x+9^x+12^x+4^x+----$
the remaining terms are omitted

wrong

to give an example $3^{.5} = 1.7 < 2^{.5} + 1^{.5 }$ and not > (given approximately)
 
  • #6
kaliprasad said:
wrong

to give an example $3^{.5} = 1.7 < 2^{.5} + 1^{.5 }$ and not > (given approximately)
let $f(x)=1+8^x+27^x-----(1)$
$g(x)=2^x+9^x+12^x----_(2)$
if $x>0$ then (1)>(2)
if $x=0$ then (1)=(2)=3
if $x<0$ and $ x \rightarrow -\infty$
then $f(x)\rightarrow 1$,and $g(x)\rightarrow 0$
here $f(x)$ and $g(x)$ both are increasing on $(-\infty , \infty)$
 

FAQ: No. of real solution of exponential equation

How do you determine the number of real solutions for an exponential equation?

The number of real solutions for an exponential equation can be determined by looking at the base and exponent of the equation. If the base is positive and the exponent is an even number, there will be two real solutions. If the base is negative, there will be no real solutions. If the exponent is an odd number, there will be one real solution.

Are there any other factors that affect the number of real solutions for an exponential equation?

Yes, the presence of any constants or variables in the equation can also affect the number of real solutions. If the equation has any additional terms, it may have more or less real solutions depending on the values of those terms.

Is it possible for an exponential equation to have no real solutions?

Yes, if the base of the equation is negative, it will have no real solutions. This is because raising a negative number to any power will always result in a negative number, and there is no real number that can be raised to a power to give a negative result.

Can an exponential equation have more than two real solutions?

Yes, if the exponent of the equation is a higher even number, such as 4 or 6, there can be 4 or 6 real solutions respectively. However, if the equation has any additional terms, it may have more or less real solutions depending on the values of those terms.

How can you solve an exponential equation to find the real solutions?

The most common method for solving an exponential equation is by using logarithms. By taking the logarithm of both sides of the equation, the exponent can be brought down and the equation can be solved for the variable. However, it is important to remember that not all exponential equations have real solutions, so it is important to check the conditions for the number of real solutions before solving.

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