Logarithmic Functions as Inverses
Master the fundamental concepts of logarithms, their inverse relationship with exponents, and evaluation techniques.
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Questions Covered in This Set
10 cards to master
What is the inverse relationship between exponentials and logarithms?
If bˣ = y, then log_b(y) = x. Logarithms undo exponentiation just as subtraction undoes addition.
Evaluate: log₂(8)
3, because 2³ = 8. The logarithm asks "2 to what power gives 8?"
What is the common logarithm notation?
log(x) means log₁₀(x). It's base 10 and used so frequently it doesn't require writing the base.
What is the natural logarithm notation?
ln(x) means log_e(x). It uses the special base e ≈ 2.718 and appears throughout calculus and science.
What are the domain and range of f(x) = log_b(x) where b > 1?
Domain: x > 0 (only positive numbers). Range: all real numbers. There's a vertical asymptote at x = 0.
Evaluate: log_b(1) and log_b(b) for any valid base b
log_b(1) = 0 because b⁰ = 1. log_b(b) = 1 because b¹ = b.
Convert to logarithmic form: 5³ = 125
log₅(125) = 3. The base becomes the base of the log, the exponent becomes the answer, and the result becomes the argument.
Evaluate: log₂(1/8)
-3, because 2⁻³ = 1/8. Logarithms can be negative when the argument is between 0 and 1.
How do logarithmic and exponential graphs relate?
They are reflections across the line y = x. Where y = bˣ has a horizontal asymptote, y = log_b(x) has a vertical asymptote.
Name two real-world applications of logarithms
pH scale (measuring acidity), Richter scale (earthquake magnitude), decibels (sound intensity), and information theory.
