Logarithmic Functions — AP Precalculus
1. Definition of Logarithmic Functions as Inverses ★★☆☆☆ ⏱ 3 min
A logarithmic function is the inverse of a one-to-one exponential function. Formally, for $b>0, b \neq 1$, $y = \log_b x$ if and only if $b^y = x$. Standard notation uses $\log x$ for the common logarithm (base 10) and $\ln x$ for the natural logarithm (base $e \approx 2.71828$), the most common form for calculus and continuous growth models.
Logarithmic functions reverse exponential operations, allowing us to solve for unknown exponents and model logarithmic scaling phenomena, from pH levels to decibel ratings. As the inverse of exponential functions, logarithmic functions swap the domain and range of exponential functions: logarithmic functions have a domain of only positive real numbers ($x>0$) and a range of all real numbers.
2. Core Properties and Change of Base Formula ★★☆☆☆ ⏱ 4 min
All logarithm properties are derived directly from corresponding exponent rules, due to the inverse relationship between logarithms and exponentials. The key inverse identities connecting the two function types are:
b^{\log_b x} = x \quad (x>0) \quad \text{and} \quad \log_b (b^x) = x \quad (\text{for all real } x)
- Product Rule: $\log_b (MN) = \log_b M + \log_b N$ (turns products of positive numbers into sums of logarithms)
- Quotient Rule: $\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N$ (turns quotients of positive numbers into differences of logarithms)
- Power Rule: $\log_b (M^k) = k \log_b M$ for any real $k$ and positive $M$
The change of base formula allows you to evaluate any logarithm using a standard calculator, which only computes base 10 or base $e$ logarithms:
\log_b a = \frac{\ln a}{\ln b} = \frac{\log_{10} a}{\log_{10} b} \quad (a>0, b>0, b \neq 1)
3. Solving Logarithmic Equations ★★★☆☆ ⏱ 4 min
A logarithmic equation has an unknown variable inside the argument of a logarithm. The core solution strategy uses the inverse relationship between logs and exponentials: isolate a single logarithmic term, convert to exponential form, then solve for the variable. If you have multiple logarithms on the same side, combine them using logarithm properties first.
The most critical step often missed is checking for extraneous solutions. Combining multiple logarithms erases individual domain restrictions, so all candidate solutions must be checked against the original equation, and any solution that makes a logarithm's argument non-positive must be discarded.
4. Graphing and Transformations of Logarithmic Functions ★★★☆☆ ⏱ 3 min
The parent logarithmic function $y = \log_b x$ has consistent core features: it has a vertical asymptote at $x=0$ (the y-axis), passes through $(1,0)$ (since $\log_b 1 = 0$ for any valid $b$), has domain $(0, \infty)$ and range of all real numbers. If $b>1$, the function is strictly increasing and concave down; if $0<b<1$, it is strictly decreasing and concave up.
Transformations of logarithmic functions follow the same rules as all other function transformations: for $y = a \log_b (c(x-h)) + k$, $a$ controls vertical stretch/compression/reflection, $c$ controls horizontal stretch/compression/reflection, $h$ is the horizontal shift, and $k$ is the vertical shift. To find the new vertical asymptote after shifting, simply set the argument of the logarithm equal to zero and solve for $x$, which avoids mistakes from memorizing shift directions.
Common Pitfalls
Why: Students confuse the product rule with adding arguments, misremembering that logs turn products into sums, not sums into sums.
Why: Students check solutions against the final combined equation, not the original equation, forgetting that individual terms have domain restrictions lost when combining.
Why: Students confuse the change of base formula with the quotient rule, mixing up the order of operations.
Why: Students generalize the domain rule for $\log(x)$ to $\log(x^2)$ without checking when the argument is positive.
Why: Students mix up horizontal shift direction, assuming $+2$ shifts right instead of left.