Calc Real $x$ in Exponential Equation

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In summary, the conversation discusses finding the real values of x for which the equation $3^x+6^x=4^x+5^x$ holds true. It is determined that the only solutions are x=0 and x=1, as shown through the analysis of the function $f(x) = 3^{x} - 4^{x} - 5^{x} + 6^{x}$ and its derivative. No negative or greater than 1 roots exist.
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juantheron
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Calculation of all real values of $x$ for which $3^x+6^x = 4^x+5^x$
 
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Re: exponential equation

jacks said:
Calculation of all real values of $x$ for which $3^x+6^x = 4^x+5^x$

[sp]Let's consider the function...

$\displaystyle f(x) = 3^{x} - 4^{x} - 5^{x} + 6^{x}\ (1)$

It is immediate to realize that f(x) vanishes in x=0 and x=1. The derivative in x=0 is...

$\displaystyle f' (0) = ln \frac{9}{10}< 0\ (2)$

... so that around x=0 is f(x)>0 for x<0 and vice versa. If x is negative then the term $3^{x}$ is dominating and is f(x)>0, so that no negative roots exist. For x>1 the term $6^{x}$ is dominating so that there is no roots greater than 1. The conclusion is that x=0 and x=1 are the only roots...[/sp]

Kind regards

$\chi$ $\sigma$
 

FAQ: Calc Real $x$ in Exponential Equation

What is an exponential equation?

An exponential equation is a mathematical expression that involves a variable as an exponent. It typically takes the form of y = ab^x, where a and b are constants and x is the variable.

How do I solve an exponential equation?

To solve an exponential equation, you can use the properties of logarithms or perform algebraic manipulations to isolate the variable. Alternatively, you can use a graphing calculator or online tool to find the numerical solution.

What is the difference between a linear and exponential equation?

A linear equation has a constant rate of change, while an exponential equation has a constant ratio of change. This means that the values in a linear equation increase or decrease by the same amount, while the values in an exponential equation increase or decrease by the same percentage.

Why is it important to find the real value of x in an exponential equation?

In many real-world applications, the value of x in an exponential equation represents a physical quantity, such as time or population. Therefore, finding the real value of x allows us to accurately predict and analyze these quantities in various scenarios.

What are some common applications of exponential equations?

Exponential equations are commonly used in finance and economics, biology, physics, and chemistry. They can be used to model population growth, compound interest, radioactive decay, and many other natural phenomena.

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