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[h] IB Mathematics AA HL Flashcards- Exponential & logarithmic functions
[q] Exponential & logarithmic functions
[a] EXPONENTIAL FUNCTIONS:
We have seen quadratics & polynomials, which contain exponents, but the variable is the base. In exponential functions, the variable is the exponent, such as \( f(x) = 2^x \), \( y = e^{x+1} \), or \( g(n) = 2000(1.06)^n \) — [compound int. function]. Compound interest and geometric sequences in general can be represented with an exponential function, and we know the behavior of these: As long as the base is more than 1, it grows (exponentially).
[q]
If we have \( f(x) = k(a)^x + c \) in general, the y-intercept is \( f(0) = k(a)^0 + c = k + c \rightarrow (0, k + c) \). Then, \( k(a)^x \) never equals zero, so we have a horizontal asymptote at \( y = c \). Graph grows if \( a > 1 \), falls if \( a < 1 \).
[a] LOGARITHMIC FUNCTIONS:
We’ve seen in previous sections that if we find the inverse of an exponential function, it is a logarithmic function. For example, if \( f(x) = e^x \), then \( f^{-1}(x) = \ln(x) \). Then, as we know, x & y are ‘switched’ in every sense: horizontal asymptotes become vertical, y-intercepts become x-intercepts, etc.
[q]
Furthermore, here are some log basics that could help you:
– \( \log 1 = 0 \) because \( x^0 = 1 \)
– \( \log 0 \) & \( \log (-ve) \) are undefined because \( x^n \neq 0 \) or \( x^n \neq -ve \) if \( x > 0 \)
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