Can You Solve x*log(x) = 0.1*x^2 Using Logarithmic Properties?

[mycrafish]

x*log(x)=0.1*x^2

 

[scurty]

My guess is that using Newton's Method would be the easiest. I don't see any easy way to isolate x in that equation.

 

[NascentOxygen]

x*log(x)=0.1*x^2

Is log = log₁₀? Then I'd guess about, oh, let's see, ... 10?

 

[HallsofIvy]

First, of course, x= 0 is not a solution because log(0) is not defined. So you can divide both sides by x to get log(x)= 0.1 x. You can now write this as $x= e^{0.1x}$. If you let y= -0.1x, that becomes $-y/0.1= e^{-y}$. Now multiply on both sides by $-0.1e^y$ to get $ye^y= -0.1$.

Now we can take the Lambert W function of both sides:
y= -0.1x= W(-0.1) so x= -10W(-0.1)= -10(-0.111833)= 1.11833 (to six significant figures).

(The Lambert W function is defined as the inverse function to $f(x)= xe^x$. It is also known as the "ProductLog" function. Mathematica evaluates that function and it can be evaluated at http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ProductLog. That's what I used to get the value above.)

 

[CellCoree]

To solve this equation, we first need to understand the properties of logarithms. Logarithms are the inverse operation of exponentiation, meaning that if we have a logarithm with a base of x, it can be rewritten as x^y = z, where x is the base, y is the exponent, and z is the result.

In this case, we have x*log(x) = 0.1*x^2. To solve for x, we can rewrite this equation as x^(log(x)) = 0.1*x^2. We can then use the property of logarithms that states log(x^y) = y*log(x) to rewrite the equation as x^(log(x)) = x^(0.1*x^2).

Since the bases on both sides of the equation are the same, we can set the exponents equal to each other, giving us log(x) = 0.1*x^2. We can then use the definition of logarithms, which states that log(x) = y if and only if x = 10^y, to rewrite the equation as x = 10^(0.1*x^2).

At this point, we can use a numerical method, such as graphing or using a calculator, to approximate the value of x that satisfies this equation. Alternatively, we can use a computer program or mathematical software to solve for x precisely.

In summary, to solve a logarithmic equation such as x*log(x) = 0.1*x^2, we need to understand the properties of logarithms and use them to rewrite the equation in a way that allows us to solve for x. From there, we can use numerical or computational methods to find the value of x that satisfies the equation.