Solve Exponential Equation: 5^(sqrt(x)) + (1/5)^(sqrt(x)) = 25

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In summary, an exponential equation is an equation with a variable in the form of x^n, where n is a constant. To solve an exponential equation, we must isolate the variable and use logarithms. The solution to the equation 5^(sqrt(x)) + (1/5)^(sqrt(x)) = 25 is x = 4. This equation only has one solution because the exponential function is one-to-one. It cannot be solved without using logarithms.
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scientifico
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Homework Statement


5^( \sqrt(x) ) + 5 * 5^(- \sqrt(x) ) = 25 + 1/5

2. The attempt at a solution

I have thought to transform 5^ (-\sqrt(x) ) in (1/5)^(\sqrt(x) ) but I don't know how to solve then... could you help me ?

thank you
 
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  • #2
scientifico said:

Homework Statement


5^( \sqrt(x) ) + 5 * 5^(- \sqrt(x) ) = 25 + 1/5

2. The attempt at a solution

I have thought to transform 5^ (-\sqrt(x) ) in (1/5)^(\sqrt(x) ) but I don't know how to solve then... could you help me ?

thank you

That's a good idea. So if u=5^(sqrt(x)) then 5^(-sqrt(x)) equals 1/u. Solve for u first.
 

FAQ: Solve Exponential Equation: 5^(sqrt(x)) + (1/5)^(sqrt(x)) = 25

1. What is an exponential equation?

An exponential equation is an equation in which the variable appears in an exponent. In other words, the variable is in the form of x^n, where n is a constant.

2. How do I solve an exponential equation?

To solve an exponential equation, we must isolate the variable on one side of the equation and use logarithms to solve for the variable. In this case, we can use the logarithmic property: log(a^b) = b*log(a).

3. What is the solution to the equation 5^(sqrt(x)) + (1/5)^(sqrt(x)) = 25?

The solution to this equation is x = 4. We can solve this by taking the logarithm of both sides and using the property mentioned in question 2. This will give us the equation sqrt(x)*log(5) + sqrt(x)*log(1/5) = log(25). Simplifying this, we get sqrt(x)*(log(5) + log(1/5)) = log(25). Using the property log(a) + log(b) = log(ab), we can simplify further to sqrt(x)*log(1) = log(25), which gives us sqrt(x) = log(25). Finally, solving for x, we get x = 4.

4. Can this equation have more than one solution?

No, this equation only has one solution, which is x = 4. This is because the exponential function is a one-to-one function, meaning that for every input there is only one output. Therefore, the equation can only have one solution.

5. Can this equation be solved without using logarithms?

No, this equation cannot be solved without using logarithms. Since the variable is in the form of an exponent, we need to use logarithms to isolate the variable and solve for it.

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