Rucete ✏ AP Precalculus In a Nutshell
7. Logarithmic Functions
This chapter introduces logarithmic functions as inverses of exponentials. You’ll learn how to evaluate logarithms, graph them, apply their properties, solve equations, and model data using logarithmic regression and semi-log plots.
- Logarithmic Expressions and Evaluation
- log₍b₎(c) means: “To what power must b be raised to get c?”
- Example: log₃(81) = 4 because 3⁴ = 81
- Common log: base 10 → log(x) = log₁₀(x)
- Natural log: base e → ln(x) = logₑ(x)
- Use calculator to find values like log(35) ≈ 1.5441
- Change of base: log₍b₎(c) = log(c) / log(b)
- Inverse of Exponential Functions
- log₍b₎(x) is the inverse of bˣ
- Graph is a reflection of exponential over y = x
- Domain: (0, ∞); Range: (−∞, ∞)
- x-intercept: (1, 0); no y-intercept
- Vertical asymptote at x = 0
- If b > 1 → increasing; if 0 < b < 1 → decreasing
- Properties of Logarithmic Functions
- log₍b₎(xy) = log₍b₎(x) + log₍b₎(y)
- log₍b₎(x/y) = log₍b₎(x) − log₍b₎(y)
- log₍b₎(xⁿ) = n · log₍b₎(x)
- Use these to expand or condense logs
- Transformations: shifts, dilations, reflections
- The Natural Log Function ln(x)
- ln(x) = logₑ(x), where e ≈ 2.718
- Key properties:
- ln(e) = 1
- ln(eⁿ) = n
- ln(1) = 0
- ln follows all the usual log rules
- Solving Logarithmic and Exponential Equations
- If log₍b₎(A) = log₍b₎(B), then A = B
- If bases differ, use logs to solve
- Always check domain: no log of 0 or negatives
- Solving exponential:
- Example: 2ˣ = 6 → log both sides → x = log(6)/log(2)
- Logarithmic Inequalities
- Solve by:
- Rewriting in exponential form
- Checking domain restrictions
- Testing intervals
- Example: log(x) < 2 → x < 100 but x > 0 → 0 < x < 100
- Transformations and Inverses
- f(x) = log₂(x − 1) + 4 → shift right 1, up 4
- To find inverse:
- Switch x and y; solve using logs or exponents
- ln(x) and eˣ are inverses
- Modeling with Logarithmic Functions
- Used for:
- Earthquakes (Richter scale)
- pH scale
- Diminishing returns
- Steps:
- Find equation from data
- Use graph transformations
- Apply regression tools (e.g., lnReg)
- Example: y = 2 ln(x) + 3 models rapid early growth then slows
- Semi-log Plots
- Used when y-values vary widely
- y-axis is log scale; x-axis is linear
- Exponential curves → straight lines on semi-log plots
- Used for:
- Population growth
- Radioactive decay
- Bacterial growth
- Linearizing Exponential Data
- From y = abˣ → log both sides → log(y) = log(a) + x · log(b)
- Linear form: slope = log(b), y-intercept = log(a)
- Use to estimate model parameters from data
In a Nutshell
Logarithmic functions are the inverses of exponential functions and are vital for solving equations, analyzing growth patterns, and modeling real-world data. Mastering their properties, graphs, transformations, and uses—including regression and semi-log plots—equips you to interpret complex change and connect exponential behavior to real-life situations.