Help Needed: Explaining x^{x} = e^{xlgx} and x^{a}

  • Thread starter pamparana
  • Start date
In summary, x^{x} is an exponential expression with the base and exponent both being x. Similarly, e^{xlgx} is another exponential expression with the base being the mathematical constant e and the exponent being x multiplied by the logarithm of x. When x^{x} = e^{xlgx}, it means that these two exponential expressions are equal to each other, showing a relationship between their base and exponent. To solve for x, one can take the natural logarithm of both sides and apply the power rule of logarithms. This equation has significance in representing a mathematical relationship between different exponential expressions and can be useful in solving other mathematical problems.
  • #1
pamparana
128
0
Hello,

I am having trouble getting my head around this:

Can someone explain why
[tex]x^{x}[/tex] = [tex]e^{xlgx}[/tex]

I cannot seem to understand why this is true. I am quite weak when it comes to handling exponentials. I dare say that I am terrified of e!

Also, would this also hold for a static power: so [tex]x^{a}[/tex]

Thanks,

Luca
 
Mathematics news on Phys.org
  • #2
Remember that for a > 0, [tex]\ln ( a^b ) = b \ln a[/tex] and the fact that exponentation and the natural logarithm are inverse functions.
 
  • #3
That makes sense!

Many thanks!
 

1. What is x^{x}?

X^{x} is an exponential expression where the base is x and the exponent is also x.

2. What is e^{xlgx}?

E^{xlgx} is another exponential expression where the base is the mathematical constant e and the exponent is x multiplied by the logarithm of x.

3. What does x^{x} = e^{xlgx} mean?

This equation means that the two exponential expressions, x^{x} and e^{xlgx}, are equal to each other. This shows a relationship between the base and exponent of each expression.

4. How do you solve x^{x} = e^{xlgx} for x?

This equation can be solved by taking the natural logarithm (ln) of both sides. This will result in ln(x^{x}) = xlgx. Then, apply the power rule of logarithms to get xln(x) = xlgx. Finally, divide both sides by x to get ln(x) = lgx, and solve for x using algebraic methods.

5. What is the significance of x^{x} = e^{xlgx}?

This equation represents a mathematical relationship between two different exponential expressions. It also shows how the base and exponent can be manipulated to create equivalent expressions. Understanding this relationship can be useful in solving other mathematical problems involving exponential expressions.

Similar threads

A Why the triangle inequality is greater than the 2 max{f(x),g(x)}
    • General Math
Replies
1
Views
927
I Why does the integral of sine of x^2 from - infinity to + infinity diverge?
    • General Math
Replies
4
Views
1K
I Statistical modeling and relationship between random variables
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
449
I Slope of LS line = Cov(X,Y)/Var(X). Intuitive explanation?
    • General Math
Replies
1
Views
3K
I There is no relation between e and π
    • General Math
Replies
7
Views
1K
MHB Prove $x>2y$ Given $e^y=\dfrac{x}{1-e^{-x}}$
    • General Math
Replies
7
Views
2K
B How do I invert this exponential function?
    • Calculus
Replies
3
Views
2K
Syntax Error? x and y components of E field in unit vector form
    • Introductory Physics Homework Help
Replies
13
Views
600
Chemistry Why Do Sulfurous Compounds Show Different Peaks in X-ray Absorption Spectra?
    • Biology and Chemistry Homework Help
Replies
1
Views
225
I Truncated Fourier transform and power spectral density
    • General Math
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top