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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.1.73c
Capitolo 7, Problema 7.1.73c

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.
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Graphs of Logarithmic Functions

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.
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Change of Base Property

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.
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Exponential Functions
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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

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Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.

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Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.


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ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

c. Show that ln 2ⁿ > n/2 and ln 2^(−n) < −n/2.

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