# Solving the Equation a^x=x: What is C?

• MHB
• Vali
In summary, the equation a^x=x with a>1 has one solution when a=e^(1/e) or C is the correct answer. This can be found by taking the derivative of the function and setting it equal to 0, which leads to the solution of x=1/ln(a) and a=e^(1/e).
Vali
If the equation a^x=x with a>1 has one solution then:
A)a=1/e
B)a=e
C)a=e^(1/e)
D)a=e^e
E)1/(e^e)
The right answer is C.I tried to derivate then to resolve f'(x) but didn't work

Vali said:
If the equation a^x=x with a>1 has one solution then:
A)a=1/e
B)a=e
C)a=e^(1/e)
D)a=e^e
E)1/(e^e)
The right answer is C.I tried to derivate then to resolve f'(x) but didn't work

I would write:

$$\displaystyle f(x)=a^x-x=0$$

Hence:

$$\displaystyle f'(x)=a^x\ln(a)-1=0$$

These imply:

$$\displaystyle x=\frac{1}{\ln(a)}=\log_a(e)\implies a^x=e$$

And so:

$$\displaystyle \ln(a)=\frac{1}{e}\implies a=e^{\frac{1}{e}}$$

MarkFL said:
I would write:

$$\displaystyle f(x)=a^x-x=0$$

Hence:

$$\displaystyle f'(x)=a^x\ln(a)-1=0$$

These imply:

$$\displaystyle x=\frac{1}{\ln(a)}=\log_a(e)\implies a^x=e$$

And so:

$$\displaystyle \ln(a)=\frac{1}{e}\implies a=e^{\frac{1}{e}}$$

Thank you for your response.I don't understand why x = 1/lna
from a^xlna-1=0 => a^x=1/lna; why x = 1/lna ?

Vali said:
Thank you for your response.I don't understand why x = 1/lna
from a^xlna-1=0 => a^x=1/lna; why x = 1/lna ?

The second equation implies:

$$\displaystyle a^x=\frac{1}{\ln(a)}$$

And the first equation implies:

$$\displaystyle a^x=x$$

Hence:

$$\displaystyle x=\frac{1}{\ln(a)}$$

Thank you very much for your help!

## 1. What is the value of the constant C when solving the equation a^x=x?

The value of C is dependent on the given value of a. It can be calculated by taking the natural logarithm of both sides of the equation and rearranging to solve for C.

## 2. Can there be multiple values of C for a single solution to the equation a^x=x?

Yes, there can be multiple values of C for a given solution. This is because the natural logarithm function is multivalued, meaning it can have multiple outputs for a single input.

## 3. How does the value of a affect the value of C in the equation a^x=x?

The value of a can greatly impact the value of C. Generally, as the value of a increases, the value of C decreases. However, there is no specific relationship between the two variables and it ultimately depends on the specific values of a and C in the equation.

## 4. Is it possible to have a negative value of C when solving the equation a^x=x?

No, it is not possible to have a negative value of C. This is because both sides of the equation must be positive for the equation to hold true. If C were negative, then the left side of the equation (a^x) would also be negative, making it unequal to the right side (x).

## 5. Can the equation a^x=x have more than one solution?

Yes, the equation can have multiple solutions. For example, when a=1, the equation becomes 1^x=x, which has an infinite number of solutions. In general, the equation will have one or more solutions depending on the value of a.

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