Unit 7 - Lesson 9: Exam Review

Modeling

• Exponential Growth: $y = ab^x$, where $a > 0$ and $b > 1$.
• Exponential Decay: $y = ab^x$, where $a > 0$ and $0 < b < 1$.

Equivalence

• Logarithms are exponents: $log_b a = c$ if and only if $b^c = a$.

Function

• Exponential function $y = b^x$ and logarithmic function $y = log_b x$ are inverse functions.

Exponential Models (Lesson 7-1)

• General form: $y = ab^x$, where $a ≠ 0$, $b > 0$, and $b ≠ 1$.
  • $b > 1$: exponential growth (b is the growth factor).
  • $0 < b < 1$: exponential decay (b is the decay factor).
• y-intercept: $(0, a)$.

Properties of Exponential Functions (Lesson 7-2)

• Transformations: Translated, stretched, compressed, and reflected.
• $y = ab^{x-h} + k$: Parent function $y = b^x$ stretched/compressed by $|a|$, reflected across x-axis if $a < 0$, translated h units horizontally, and k units vertically.
• Continuously Compounded Interest: $A = Pe^{rt}$, where P is principal, r is annual interest rate, and t is time in years.

Logarithmic Functions as Inverses (Lesson 7-3)

• If $x = b^y$, then $log_b x = y$. Logarithmic functions are inverses of exponential functions.
• $y = a \cdot log_b(x-h) + k$: Translations, stretches, compressions, and reflections.
• Common logarithm: base 10, written as $log x$.

Properties of Logarithms (Lesson 7-4)

• Product Property: $log<em>b mn = log</em>b m + log_b n$
• Quotient Property: $log<em>b (m/n) = log</em>b m - log_b n$
• Power Property: $log<em>b m^n = n \cdot log</em>b m$

Exponential and Logarithmic Equations (Lessons 7-5 and 7-6)

• Exponential equation form: $b^{f(x)} = a$. Solve by taking the logarithm of both sides.
• Logarithmic equations: Equations with logarithms involving a variable.

Natural Logarithms (Lesson 7-8)

• Inverse of $y = e^x$ is $y = log_e x = ln x$.