Understanding the Equality ( e^{log x^2} x^2 ) with Detailed Analysis

In this article, we will delve into the mathematics behind the equality ( e^{log x^2} x^2 ). We will explore the properties of logarithms and exponents, and understand why this equality holds true. This discussion will be helpful for anyone interested in understanding the intricacies of these mathematical functions, and it will be particularly useful for those studying calculus and logarithmic functions.

Introduction to Logarithms and Exponents

Logarithms and exponents are fundamental concepts in mathematics. Logarithms are the inverse operations of exponentiation. This means that if we have the exponential function ( b^y a ), the logarithmic function ( log_b(a) y ) is the inverse operation. In this case, we are dealing with the natural logarithm (logarithm with base ( e )), where ( e ) is the base of the natural logarithm.

Step-by-Step Breakdown of ( e^{log x^2} x^2 )

Let's break down the expression ( e^{log x^2} x^2 ) step-by-step.

Understanding the Logarithm

Logarithm is the inverse of exponentiation. Specifically, for the natural logarithm (base ( e )), ( log x ) gives the power to which ( e ) must be raised to yield ( x ). This means that ( e^{log x} x ) for any positive ( x ).

Applying the Property

When we have the expression ( e^{log x^2} ), we can think of ( x^2 ) as the argument of the logarithm. Since ( x^2 ) must be positive (because the logarithm is only defined for positive values), we can use the property of the exponential and logarithm functions:

Step 1: Start with ( e^{log x^2} ).

Step 2: By the definition of logarithms, ( log x^2 ) gives the power to which ( e ) must be raised to obtain ( x^2 ). Therefore, when we take ( e ) to that power, we simply return to ( x^2 ).

Conclusion

Thus, the equality ( e^{log x^2} x^2 ) holds true for any positive value of ( x ).

Note: This equality is only true if the logarithm is of base ( e ) since according to the rules of logarithms, ( log_a x 1 ) if ( a x ).

Further Explanation with Examples

Let's consider a more general case where we multiply the logarithm on both sides of the equation:

Given: ( e^{log x^2} x^2 )

Multiplying both sides with log (base 10) (denoted as log):

( log(e^{log x^2}) log(x^2) )

Since ( log(e) 1 ) (not to be confused with ( ln(e) 1 )), the equation simplifies to:

( (log x^2) (log e) log x^2 )

( (log x^2) (1) log x^2 )

( log x^2 log x^2 )

This confirms that the equality holds true under these conditions.

Using Y as a Variable

Let's express this using the variable ( y ):

Let ( y e^{log x^2} ). Taking the natural logarithm (base ( e )) on both sides:

( log y log(e^{log x^2}) )

( log y (log x^2) (log e) )

Since ( log e 1 ):

( log y log x^2 )

Exponentiating both sides with base ( e ):

( y x^2 )

This shows that the equality ( e^{log x^2} x^2 ) is indeed true.

Conclusion

In summary, the equality ( e^{log x^2} x^2 ) is a direct consequence of the properties of logarithms and exponents. It is crucial to understand that this equality holds only when the logarithm is of base ( e ). This understanding is essential for anyone studying logarithmic functions and their applications in various fields of mathematics and science.