- #1
Little ant
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Find the solutions of 5^x=2x+1 by ordinary methods? if it can´t be found by these ways, then prove that it's imposible.
Solve 5^x=2x+1 or Prove Impossibility
- Thread starter Little ant
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In summary, finding the solutions of 5^x=2x+1 by ordinary methods depends on the definition of "ordinary methods." By inspection, one solution is x=0, but there is another solution that can be found by studying the function y=(5^x)-(2x+1) and using numerical methods such as Newton-Raphson. However, analytical solving using the Lambert W function is not considered an ordinary method.
- #2
stevenb
- 701
- 7
It depends on what you mean by ordinary methods.
By inspection, x=0.
By inspection, x=0.
- #3
Little ant
- 21
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but, there is other answer, how you find it?
- #4
Femme_physics
Gold Member
- 2,550
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The exponent ^5 is behind the X or after?
- #5
JJacquelin
- 801
- 35
Ordinary method consists in studying the function y=(5^x)-(2x+1)
This shows that two roots exist (one of them is obvious).
Drawing the graph of the function allows to obtain a first approximate of the second root.
Then, numerical computation leads to the value of the root, as accurate as we want. There are a lot of numerical methods : Newton-Raphson and many other...
Analytical solving is outside the scope of ordinary methods. It requires the use of a special function : the Lambert W function.
This shows that two roots exist (one of them is obvious).
Drawing the graph of the function allows to obtain a first approximate of the second root.
Then, numerical computation leads to the value of the root, as accurate as we want. There are a lot of numerical methods : Newton-Raphson and many other...
Analytical solving is outside the scope of ordinary methods. It requires the use of a special function : the Lambert W function.
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FAQ: Solve 5^x=2x+1 or Prove Impossibility
What is the equation 5^x=2x+1?
The equation 5^x=2x+1 is a mathematical equation where x is an unknown variable and the goal is to find the value of x that satisfies the equation. It is an exponential equation that involves a base of 5 and a linear term with a coefficient of 2 and a constant of 1.
How do you solve 5^x=2x+1?
To solve 5^x=2x+1, you can use various methods such as graphing, algebraic manipulation, or numerical approximation. One common method is to take the logarithm of both sides and use logarithmic properties to solve for x. Another approach is to use a calculator or software to plot the two equations and find their intersection point, which is the solution for x.
Is it possible to solve 5^x=2x+1?
Yes, it is possible to solve 5^x=2x+1. However, the solution may not be a real number. In some cases, the solution may be a complex number or an irrational number that cannot be expressed in decimal form. This is why numerical approximation methods are often used to find an approximate solution.
Can you prove the impossibility of solving 5^x=2x+1?
Yes, it is possible to prove the impossibility of solving 5^x=2x+1. One way to do this is by using mathematical techniques such as the intermediate value theorem or the mean value theorem to show that the two equations do not intersect, meaning there is no solution for x. Another approach is to prove the contradiction of the equation, such as by showing that the left side is always greater than the right side for all values of x.
Are there any real-life applications of the equation 5^x=2x+1?
The equation 5^x=2x+1 is commonly used in mathematical modeling, specifically in exponential growth and decay problems. It can also be applied in various fields such as finance, biology, physics, and engineering to model real-world situations where quantities change exponentially. For example, in finance, this equation can be used to model the growth of investments with compound interest. In biology, it can be used to model the growth of bacteria or populations. In physics, it can be used to model the decay of radioactive substances.