Precalculus · Unit 2: Exponential and Logarithmic Functions · 16 min read · Updated 2026-05-11
Exponential and Logarithmic Functions — AP Precalculus
AP Precalculus · Unit 2: Exponential and Logarithmic Functions · 16 min read
1. Exponential Growth and Decay★★☆☆☆⏱ 3 min
In the general form $f(x) = ab^x$, $a$ is the non-zero initial value (output at $x=0$), and $b>0, b \neq 1$ is the base. If $b>1$, it is exponential growth; if $0 < b < 1$, it is exponential decay. For continuous change, we use the natural exponential form:
A(t) = A_0 e^{kt}
Where $A_0$ is initial amount, $k$ is the continuous rate (positive for growth, negative for decay), and $t$ is time. All exponential functions have domain $(-\infty, \infty)$, range $(0, \infty)$, and a horizontal asymptote at $y=0$.
2. Logarithmic Functions and Properties★★★☆☆⏱ 4 min
Logarithmic functions have domain $(0, \infty)$, range all real numbers, and a vertical asymptote at $x=0$. All core logarithm properties are derived from exponent rules, required for the AP exam:
**Product Rule**: $\log_b(xy) = \log_b x + \log_b y$
**Quotient Rule**: $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$
**Power Rule**: $\log_b(x^p) = p\log_b x$
**Change of Base Formula**: $\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}$
**Inverse Identities**: $b^{\log_b x} = x$ and $\log_b(b^x) = x$
3. Solving Exponential and Logarithmic Equations★★★☆☆⏱ 4 min
For exponential equations: if both sides can be rewritten with the same base, set exponents equal. If not, take the logarithm of both sides, use the power rule to bring down the exponent, then solve. For logarithmic equations, follow these steps:
Isolate the logarithmic term on one side of the equation
Convert to equivalent exponential form
Solve for the unknown variable
Check candidate solutions against the domain (log arguments must be positive) and discard extraneous solutions
4. Modelling with Exponential and Logarithmic Functions★★★★☆⏱ 3 min
Exponential and logarithmic functions model real-world scenarios where change is proportional to the current quantity, including population growth, radioactive decay, compound interest, pH levels, and earthquake magnitude. A common tested context is compound interest:
**Discrete compounding** (fixed intervals: monthly/quarterly): $A = P\left(1 + \frac{r}{n}\right)^{nt}$, where $P$ = principal, $r$ = annual rate, $n$ = compounding periods per year, $t$ = time in years
**Continuous compounding**: $A = Pe^{rt}$, the limit of discrete compounding as $n \to \infty$
5. AP-Style Concept Check★★★★☆⏱ 2 min
Common Pitfalls
Why: Students focus on algebraic manipulation and ignore the strict domain restriction that log arguments must be positive
Why: Students memorize formulas without matching them to the problem context
Why: Students overgeneralize the product and power rules to non-qualifying expressions
Why: Students mix up the properties of a function and its inverse