DPST1013 Chap.10 Logarithmic and Hyperbolic Functions
Logarithmic and Hyperbolic Functions
The natural logarithm is introduced through the integral of one over x and inverted by the exponential function. Log laws simplify products and powers only on suitable domains. Hyperbolic sine and cosine recombine exponentials, producing parallel derivative rules and a fundamental identity with a minus sign.
What this chapter covers
- 01
Natural logarithm
- 02
Exponential inverse
- 03
Log laws and domains
- 04
Logarithmic differentiation
- 05
Hyperbolic functions
- 06
Inverse hyperbolic functions and substitutions
Worked example · free
Solve an exponential equation exactly
- 1Isolate the exponential to get e to the power 2x equals 7.
- 1Take natural logarithms to obtain 2x = ln 7.
- 1Divide by 2, giving x = (ln 7)/2.
- 1Substitution gives 3 times 7 minus 5 = 16.
Key terms
- Natural logarithm
- The inverse of the natural exponential function on positive real inputs.
- Logarithmic differentiation
- A method that takes logarithms before differentiating products, quotients or variable powers.
- Hyperbolic cosine
- The even function formed by averaging e to x and e to negative x.
- Hyperbolic sine
- The odd function formed by half the difference of e to x and e to negative x.
Logarithmic and Hyperbolic Functions FAQ
Why do logarithm laws need domain conditions?
Real logarithms accept only positive arguments. Expanding or combining logs without checking positivity can create expressions on a larger or smaller domain than the original. Absolute values are needed in several antiderivative identities.
Is inverse hyperbolic sine a reciprocal?
No. The superscript minus one denotes function inversion in this context. The reciprocal of hyperbolic sine is cosecant hyperbolic, written csch. Always distinguish inverse notation from reciprocal notation before manipulating an expression.
How can I recover a forgotten hyperbolic identity?
Return to the exponential definitions of sinh and cosh, expand and simplify. This reconstructs the identity cosh squared minus sinh squared equals one and prevents importing the wrong sign from trigonometric identities.
How should I solve an equation with the variable in an exponent?
Isolate the exponential expression, confirm its required positivity and take a logarithm on both sides. Use the power rule to bring the exponent down, solve the resulting algebraic equation and substitute back. If multiple exponential terms occur, a substitution may turn the equation into a polynomial. Numerical methods may be necessary when the variable remains in incompatible positions.
Why does logarithmic differentiation help with variable powers?
Taking a logarithm converts products into sums and brings variable exponents down as factors, where product and chain rules can handle them. Differentiate implicitly, then multiply by the original function to recover its derivative. Record domain assumptions or use an appropriate absolute-value formulation. This technique organises expressions such as a function raised to another function without expanding impossible products.
What are the key monotonicity differences between sinh and cosh?
Hyperbolic sine has derivative cosh, which is always positive, so it is strictly increasing and invertible on the whole real line. Hyperbolic cosine has derivative sinh and decreases before zero then increases after zero, so it is not one-to-one on the whole line. An inverse cosh therefore uses a restricted domain, conventionally the non-negative half-line.
How do inverse hyperbolic formulas lead to logarithms?
Start with the exponential definition, introduce a temporary exponential variable and solve the resulting quadratic. Positivity of an exponential selects the admissible root. Taking the natural logarithm then isolates the original variable. The square-root and domain conditions differ among inverse sinh, cosh and tanh, so derive or quote each formula together with its valid input range.
What checks should follow a logarithmic simplification?
Compare the domains of the original and simplified expressions, since algebraic-looking log laws are valid only where their arguments meet the real logarithm conditions. Test by re-expanding products, quotients and powers, and substitute a permissible numerical value. If an absolute value appears in an antiderivative or identity, do not discard it unless a stated interval fixes the sign.
Exam move
Keep a domain line beside every logarithmic manipulation. Solve exponential equations by isolating the exponential, taking logs and checking in the original equation. For variable powers, use logarithmic differentiation and multiply back by the original function. Rebuild hyperbolic identities from exponential definitions when uncertain, and use the positivity of cosh to choose square-root signs.
Check inverse notation carefully before treating any expression as a reciprocal. Organise practice into transformations, equations, derivatives and inverse functions, because the same symbols invite different operations. Draw small graphs of exponential, logarithmic, sinh and cosh functions to retain range and monotonicity information.
When proving an identity, work from the more complicated side and return to exponential definitions only as far as needed. Add a domain audit after every algebraic solution, then verify surviving candidates in the unsimplified equation.
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