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
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.
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).
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.
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.
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.