Logarithmic Functions: Inverse of Exponential Functions

One-to-one and inverse relationship

  • A function y = f(x) is one-to-one if different x produce different y (horizontal line test). Such a function has an inverse f^{-1} defined implicitly by the equation x = f(y) (equivalently y = f^{-1}(x)).
  • The exponential function y = a^x, with base a > 0 and a ≠ 1, is one-to-one, so it has an inverse.

Logarithmic function: Definition and key properties

  • The logarithmic function with base a is denoted by y = \log_a x, where a > 0 and a ≠ 1.
  • It is defined by the equivalence: y=log⁡ax  ⟺  x=ay.y = \log_a x \iff x = a^y.
  • Domain: x > 0.
  • Since the range of y = a^x is (0, ∞), the domain of loga x is (0, ∞) and the range of loga x is all real numbers.
  • Mnemonic: a logarithm is the exponent; it tells you the exponent to which a must be raised to obtain x.
  • Base restrictions reiterated: a > 0, a ≠ 1.

Relationship to exponentials

  • The logarithm is the inverse function of the exponential; graphs of y = a^x and y = \log_a x are reflections across the line y = x.
  • Core equivalence: y=log⁡ax  ⟺  x=ay.y = \log_a x \iff x = a^y.

Illustrative examples: relating logs and exponents

  • Example: If y = \log7 x, then x = 7^y. In particular, log⁡</em>749=2  ⟺  49=72.\log</em>7 49 = 2 \iff 49 = 7^2.
  • Example: If y = \log4 x, then x = 4^y. In particular, log⁡</em>4(14)=−1  ⟺  14=4−1.\log</em>4 \left( \frac{1}{4} \right) = -1 \iff \frac{1}{4} = 4^{-1}.

Converting between exponential form and logarithmic form

  • Core rule: y=log⁡ax  ⟺  x=ay,a>0, a≠1.y = \log_a x \iff x = a^y, \quad a > 0, \ a \neq 1.
  • Try conversions from the transcript:
    • If w=1.63,w = 1.6^3, then 3=log⁡1.6w.3 = \log_{1.6} w.
    • If eu=25,e^u = 25, then u=log⁡e25.u = \log_e 25. (Natural log: ln 25)
    • If b5=27,b^5 = 27, then 5=log⁡b27.5 = \log_b 27.

Converting logarithmic statements to exponential statements

  • Core rule: If ( \log_a x = y), then x=ay.x = a^y.
  • Examples:
    • If log⁡b8=3,\log_b 8 = 3, then b3=8.b^3 = 8.
    • If log⁡3c=−2,\log_3 c = -2, then c=3−2=19.c = 3^{-2} = \frac{1}{9}.
    • If log⁡46=w,\log_4 6 = w, then 4w=6.4^w = 6.

Quick recap and practical notes

  • Remember: a log answers the question "to what exponent must the base be raised to obtain x?"
  • Domains and ranges:
    • Domain of log⁡ax\log_a x: x > 0
    • Range of log⁡ax\log_a x: all real numbers
    • Domain of axa^x: all real x
    • Range of axa^x: (0, ∞)
  • Base conditions: a > 0 and a ≠ 1
  • Natural logarithm: ln x = \log_e x