Exponential and Logarithmic Functions
Outcome 2: Exponential and Logarithmic Functions · Blueprint Pillar 3 · PGCC MAT-1350 (interim) · Download .docx
Objectives
- Identify and graph exponential growth and decay functions.
- Evaluate and simplify logarithmic expressions using properties.
- Solve exponential equations by taking logarithms of both sides.
- Solve logarithmic equations by converting to exponential form.
- Write and interpret exponential models for real-world growth and decay.
Key terms
- exponential function
- f(x) = abˣ where b > 0, b ≠ 1; models processes that grow or decay by a constant multiplicative factor.
- base b > 1
- Exponential growth — the function increases as x increases.
- base 0 < b < 1
- Exponential decay — the function decreases as x increases.
- natural exponential
- f(x) = eˣ where e ≈ 2.718; arises in continuous compound interest and natural growth.
- logarithm
- logb(y) = x means bˣ = y; the inverse of the exponential function base b.
- product rule
- logb(MN) = logb(M) + logb(N).
- quotient rule
- logb(M/N) = logb(M) − logb(N).
- power rule
- logb(Mⁿ) = n · logb(M).
- change of base
- logb(x) = ln(x)/ln(b) — converts to natural log for computation.
- continuous compounding
- A(t) = Pe^(rt) — amount after time t at continuous rate r.
The concept
Exponential functions describe processes that change by multiplication rather than addition. Linear growth adds a constant each step: 5, 10, 15, 20. Exponential growth multiplies each step: 5, 10, 20, 40. The difference is dramatic over time — exponential processes start slowly but quickly dwarf linear ones. This is why pandemic spread, compound interest, and viral content follow exponential models.
The general form is f(x) = abˣ. The coefficient a is the initial value (f(0) = a). The base b is the growth factor. If b > 1, the function grows. If 0 < b < 1, the function decays. The natural exponential f(x) = eˣ uses Euler's number e ≈ 2.71828, which arises naturally when growth is continuous.
Logarithms are inverses of exponentials. logb(y) = x means 'b to the power x gives y.' This definition is the key to solving exponential equations: if 3ˣ = 81, write log₃(81) = x. Since 3⁴ = 81, x = 4. For bases without neat integer solutions, use the change-of-base formula: x = log(81)/log(3) = 1.908/0.477 ≈ 4. Calculators evaluate log (base 10) and ln (base e) directly.
Three logarithm properties are essential: the product rule (log of a product = sum of logs), the quotient rule (log of a quotient = difference of logs), and the power rule (log of a power = exponent times log). These properties mirror exponent rules — logs of multiplication become addition, logs of division become subtraction, logs of powers become multiplication.
Solving exponential equations: isolate the base, take log of both sides, use the power rule to bring the exponent down as a coefficient, solve for the variable. Solving logarithmic equations: isolate the log, convert to exponential form (rewrite logb(x) = c as bᶜ = x), then solve for the variable. Always check that the solution makes the argument of the log positive.
Blueprint Pillar 3 — Technology and Society: Exponential and logarithmic models appear in every STEM field. Richter scale (earthquakes), pH scale (chemistry), decibel scale (sound) — all use logarithms to compress wide ranges into manageable numbers. Compound interest, radioactive half-life, drug concentration in the body, population ecology — all use exponential models. Understanding these functions is essential for interpreting quantitative information in science, business, and public health.
Worked examples
Common mistakes
- Confusing log(M + N) with log(M) + log(N). The product rule applies to a product inside the log, not a sum. log(M + N) has no simplification; log(M · N) = log M + log N.
- Forgetting domain restrictions when solving log equations. If solving log(x − 2) = 3, the solution gives x = 1002 + 2 = 1004 — but always verify x − 2 > 0, so x > 2. x = 1004 > 2. ✓
- Taking log of a sum before isolating the exponential. 3 + 2ˣ = 11 must first become 2ˣ = 8, then x = 3. Taking log of both sides before isolating gives log(3 + 2ˣ) = log(11) — this is not helpful.
Self-check
Try each one before you look. A miss here costs nothing and tells you exactly what to reread.
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