Using the new definition of lnx as an integral and e^x as...

Using the new definition of $\ln x$ as an integral and $e^{x}$ as the inverse of $\ln x$ Prove that $\ln x^{a}=a \ln x$ for any $x>0$ and any $a$ by following these steps: (a) Use Theorem 4.35 to show that $\frac{d}{d x}\left(\ln x^{a}\right)=\frac{a}{x}$. (b) Compare the derivatives of $\ln x^{a}$ and $a \ln x$ to argue that $\ln x^{a}=a \ln x+C$ (c) Use part (b) with $x=1$ and $a=1$ to show that $C=0$, and then complete the proof.

Using the new definition of $\ln x$ as an integral and $e^{x}$ as the inverse of $\ln x$
Prove that $\ln x^{a}=a \ln x$ for any $x>0$ and any $a$ by following these steps:
(a) Use Theorem 4.35 to show that $\frac{d}{d x}\left(\ln x^{a}\right)=\frac{a}{x}$.
(b) Compare the derivatives of $\ln x^{a}$ and $a \ln x$ to argue that $\ln x^{a}=a \ln x+C$
(c) Use part (b) with $x=1$ and $a=1$ to show that $C=0$, and then complete the proof.

Key Concepts

Constant of Integration and Use of Initial Conditions

When two functions have the same derivative, they differ by at most a constant. Establishing this constant with an initial condition (such as evaluating at x = 1 where ln 1 = 0) is a key step in many proofs, ensuring that the two functions are indeed equal. This method validates logarithmic identities by linking the derivative information to the specific function values.

Chain Rule in Differentiation

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When differentiating a function like ln(x^a), the chain rule allows us to differentiate the outer logarithmic function and multiply by the derivative of the inner power function, leading to the derivative a/x.

Integral Definition of the Natural Logarithm

This concept establishes the natural logarithm ln x as the definite integral from 1 to x of 1/t dt. It provides a rigorous, analytic foundation for logarithms and directly leads to its properties, such as its derivative, without resorting to series expansions or limits.

Exponential Function as the Inverse of the Logarithm

Defining the exponential function e^x as the inverse of the logarithmic function underscores the deep connection between these two functions. This relationship is critical for understanding the behavior of logarithms and exponentials, and it is instrumental in transferring properties between the two functions.

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