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

Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problema 7.1.5

Capitolo 7, Problema 7.1.5

Express 3ˣ, x^{π}, and x^{sin x} using the base e.

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Recall that any exponential expression with base \(a\) can be rewritten using the natural exponential function \(e\) as \(a^{b} = e^{b \ln a}\). This is because \(e^{\ln a} = a\).

For the expression \(3^{x}\), apply the formula by setting \(a = 3\) and \(b = x\). This gives \(3^{x} = e^{x \ln 3}\).

For the expression \(x^{\pi}\), treat \(x\) as the base and \(\pi\) as the exponent. Using the same formula, \(x^{\pi} = e^{\pi \ln x}\).

For the expression \(x^{\sin x}\), the exponent is a function of \(x\). Apply the formula with \(a = x\) and \(b = \sin x\), resulting in \(x^{\sin x} = e^{\sin x \cdot \ln x}\).

Summarize the results: \(3^{x} = e^{x \ln 3}\), \(x^{\pi} = e^{\pi \ln x}\), and \(x^{\sin x} = e^{\sin x \ln x}\). These forms express the original functions using the base \(e\).

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Exponential Functions and the Natural Base e

Exponential functions with base e are expressed as e raised to a power. The number e (~2.718) is the natural base for logarithms and exponentials, simplifying many calculus operations. Any exponential expression a^x can be rewritten as e^(x ln a), linking arbitrary bases to base e.

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Integrals of Natural Exponential Functions (e^x)

Natural Logarithm (ln) and its Properties

The natural logarithm ln(x) is the inverse of the exponential function e^x. It allows conversion between different exponential bases by using the identity a^x = e^(x ln a). Understanding ln is essential for rewriting expressions with bases other than e.

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Derivative of the Natural Logarithmic Function

Handling Variable Exponents in Exponential Expressions

When the exponent itself is a function of x, such as x^{sin x}, rewriting with base e involves expressing the power as e raised to the product of the exponent function and the natural logarithm of the base. This approach generalizes to variable exponents and is key for differentiation and integration.

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Simplifying Exponential Expressions

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Uranium dating Uranium-238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of U-238 and lead, and they determine that 85% of the original U-238 remains; the other 15% has decayed into lead. How old is the rock?

142

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29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₀ˡⁿ ² (e^{3x} − e^{−3x}) / (e^{3x} + e^{−3x}) dx

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What is the inverse function of ln x, and what are its domain and range?

262

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Arc length Use the result of Exercise 108 to find the arc length of the curve: f(x) = ln |tanh(x / 2)| on [ln 2, ln 8].

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37–56. Integrals Evaluate each integral.

∫ sinh x / (1 + cosh x) dx

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13–14. Absolute and relative growth rates Two functions f and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant.

f(t) = 100 + 10.5t, g(t) = 100e^(t/10)

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