Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:

c. (exp(x))ᵖ = exp(px), p rational

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Recall that the exponential function \( \exp(x) \) and the natural logarithm \( \ln(x) \) are inverse functions, meaning \( \ln(\exp(x)) = x \) for all real \( x \), and \( \exp(\ln(x)) = x \) for all \( x > 0 \).

Start with the expression \( (\exp(x))^{p} \), where \( p \) is a rational number. We want to rewrite this expression using the properties of logarithms and exponentials.

Apply the natural logarithm to \( (\exp(x))^{p} \) to use the logarithm power rule: \[ \ln\left((\exp(x))^{p}\right) = p \cdot \ln(\exp(x)) \].

Since \( \ln(\exp(x)) = x \), substitute this into the equation to get \[ \ln\left((\exp(x))^{p}\right) = p \cdot x \].

Now, exponentiate both sides to remove the logarithm: \[ (\exp(x))^{p} = \exp(p \cdot x) \]. This completes the proof of the property.

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Inverse Relationship Between Exponential and Logarithmic Functions

The exponential function exp(x) and the natural logarithm ln(x) are inverses, meaning exp(ln(x)) = x for x > 0 and ln(exp(x)) = x for all real x. This relationship allows us to switch between the two functions to simplify expressions and prove properties.

Properties of the Natural Logarithm

The natural logarithm has key properties such as ln(a^b) = b ln(a) and ln(ab) = ln(a) + ln(b). These properties are essential for manipulating expressions involving powers and products, which help in proving identities involving exponentials.

Exponentiation of the Exponential Function

Raising exp(x) to a rational power p means (exp(x))^p = exp(x)^p. Using the inverse and logarithm properties, this can be rewritten as exp(p x), showing how powers distribute over the exponential function when p is rational.

Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:c. (exp(x))ᵖ = exp(px), p rational