MATH1131 Chap.12 Logarithms, Exponentials and Hyperbolic Functions
Logarithms, Exponentials and Hyperbolic Functions
The course defines the natural logarithm as the integral of 1/t from 1 to x for positive x, so its derivative 1/x comes from the first fundamental theorem and its laws are proved from that definition. Because ln is strictly increasing with range all real numbers, it has an inverse, the exponential function, and the inverse function theorem shows that e^x is its own derivative.
General powers are defined by a^x = e^{x ln a}, which gives their derivatives and explains logarithmic differentiation for expressions such as x^x where both base and exponent vary. Limits of the forms one to the infinity, zero to the zero and infinity to the zero are handled by taking logarithms and then applying l'Hôpital's rule.
The hyperbolic functions cosh and sinh are built from e^x and e^{-x}; they satisfy identities that mirror the circular ones with sign changes, such as cosh squared minus sinh squared equal to one. Their inverses can be written as logarithms, and they supply antiderivatives for expressions involving the square root of x squared plus or minus a constant.
What this chapter covers
- 01
The natural logarithm defined as an integral
- 02
Laws of logarithms
- 03
The exponential function as an inverse
- 04
General powers and logarithms to other bases
- 05
Integrals giving ln of the absolute value
- 06
Logarithmic differentiation
- 07
Indeterminate forms with powers
- 08
Hyperbolic functions, identities and inverses
Worked example · free
Logarithmic differentiation of a variable power
- 1Take logarithms: ln y = sin x ln x.
- 1Differentiate implicitly using the product rule: y'/y = cos x ln x + (sin x)/x.
- 1Multiply by y: y' = x^{sin x}(cos x ln x + (sin x)/x).
Key terms
- Natural logarithm
- The function defined for positive x as the integral of 1/t from 1 to x.
- Exponential function
- The inverse of the natural logarithm, whose derivative equals itself.
- Hyperbolic cosine
- The even function cosh x, equal to the average of e to the x and e to the minus x.
- Hyperbolic sine
- The odd function sinh x, equal to half of e to the x minus e to the minus x.
- Logarithmic differentiation
- Differentiating the logarithm of a function implicitly to find the derivative of a product or variable power.
Logarithms, Exponentials and Hyperbolic Functions FAQ
Why does the course define ln as an integral?
The definition makes the derivative of ln immediate through the first fundamental theorem and allows every logarithm law to be proved rigorously. The exponential is then defined as its inverse.
How do I evaluate a limit of the form one to the infinity?
Let y be the expression, take ln to turn the power into a product, rewrite it as a quotient, and evaluate with l'Hôpital's rule. Then exponentiate the result, using continuity of the exponential function.
What is the hyperbolic version of the Pythagorean identity?
cosh squared x minus sinh squared x equals one for every real x. Note the minus sign, which differs from the circular identity cos squared plus sin squared equals one.
How is the inverse hyperbolic sine written with a logarithm?
It equals ln of x plus the square root of x squared plus one, found by solving the definition of sinh as a quadratic in e to the y and taking the positive root.
How do I differentiate a to the power x for a positive constant a?
Write it as e to the x ln a and apply the chain rule. The derivative is a to the x times ln a, which reduces to the familiar rule for e to the x because ln e equals one.
Exam move
Rebuild the main results from the definitions at least once: the derivative of ln from the fundamental theorem, the derivative of e^x from the inverse function theorem, and the logarithmic form of the inverse hyperbolic sine from a quadratic. Keep a side-by-side table of circular and hyperbolic identities and mark every sign difference.
Practise logarithmic differentiation on variable powers and limits of power forms, writing out the ln step each time. Finish by integrating expressions with square roots of x squared plus or minus a constant using hyperbolic substitutions.
Verify hyperbolic identities numerically at one value such as x equal to one using the exponential definitions, which builds confidence in the signs before you use an identity in a longer argument.
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