Exponential and Logarithmic Functions by Owen Borville April 13, 2026
Exponential and logarithmic functions model processes where change is proportional to the current amount (exponential) or where you need to undo that proportional change (logarithm). These functions appear across finance, natural sciences, engineering, and data analysis.
Quick comparison of common applications
Application-Typical model-Key parameter-Concrete example:
Compound interest A=P(1+r/n)^nt or A=Pe^rt; r is the annual rate; Growth of savings account balance.
Population growth P(t)=P0e^kt; k is the relative growth rate; Bacterial culture or human population projection.
Radioactive decay N(t)=N0e^−kt or half‑life t1/2=ln2/k; t₁/₂ is the half‑life; Decay of isotopes in dating or medicine.
Newton’s Law of Cooling T(t)=Tenv+(T0−Tenv)e^−kt; k is the cooling constant; Cooling of a hot object in room temperature.
Logistic growth P(t)=L/(1+Ae^−kt); L is the carrying capacity; Population with limited resources or spread of ideas.
Logarithmic scales y=logb(x); base b; pH, Richter magnitude, decibel scales compress wide ranges.
Disease spread (in epidemiology) and viruses are commonly modeled using exponential and logarithmic models. In the early stages, epidemics grow exponentially, which presents as a straight line on a semi-logarithmic scale (log of cases vs. time), allowing for the estimation of growth rates. As infections spread, the pool of susceptible individuals depletes, leading to a deceleration in growth that can be modeled with logarithmic, logistic, or SIR models.
Logarithms in chemical kinetics are used to linearize exponential data, allowing for the determination of rate constants, activation energies, and reaction orders.
How to set up and solve exponential and logarithmic functions and equations:
Compound interest and continuous growth:
Model: A=Pe^rt for continuous compounding
Solve for time when you want to reach amount A: take natural log to get t=ln(A/P)/r
Use: planning investments or loans.
Decay and half‑life:
Model: N(t)=N0e^−kt.
Find half‑life: set N(t)=1/2N0 and solve t1/2=ln2/k.
Use: radiocarbon dating, pharmacokinetics.
Inverse problems with logarithms:
Question type: “How long until X reaches Y?”
Method: take ln of both sides of an exponential model and solve for the time variable.
Use: time to double an investment, time for cooling to a target temperature.
Example 1: Compound interest
Problem: $2,000 invested at 4% compounded continuously. How long to reach $3,000? Solution: 3000=2000e^0.04t. Take ln: t=ln(1.5)/0.04≈0.4055/0.04=10.14 years.
Example 2: Radioactive decay half‑life
Problem: A sample decays to 30% of its original amount in 5 years. Find half‑life. Solution: 0.30=e^−5k so k=−ln0.30/5. Then t1/2=ln2/k. Compute numerically for the half‑life.
Tips for modeling and interpretation: Choose the right model: exponential for unconstrained proportional change, logistic when there’s a limiting capacity. Check units: ensure the rate r or k matches the time units of t. Use logarithms to linearize: plotting ln(y) vs t turns y=Ae^kt into a straight line with slope k, useful for parameter estimation.
Exponential and logarithmic functions model processes where change is proportional to the current amount (exponential) or where you need to undo that proportional change (logarithm). These functions appear across finance, natural sciences, engineering, and data analysis.
Quick comparison of common applications
Application-Typical model-Key parameter-Concrete example:
Compound interest A=P(1+r/n)^nt or A=Pe^rt; r is the annual rate; Growth of savings account balance.
Population growth P(t)=P0e^kt; k is the relative growth rate; Bacterial culture or human population projection.
Radioactive decay N(t)=N0e^−kt or half‑life t1/2=ln2/k; t₁/₂ is the half‑life; Decay of isotopes in dating or medicine.
Newton’s Law of Cooling T(t)=Tenv+(T0−Tenv)e^−kt; k is the cooling constant; Cooling of a hot object in room temperature.
Logistic growth P(t)=L/(1+Ae^−kt); L is the carrying capacity; Population with limited resources or spread of ideas.
Logarithmic scales y=logb(x); base b; pH, Richter magnitude, decibel scales compress wide ranges.
Disease spread (in epidemiology) and viruses are commonly modeled using exponential and logarithmic models. In the early stages, epidemics grow exponentially, which presents as a straight line on a semi-logarithmic scale (log of cases vs. time), allowing for the estimation of growth rates. As infections spread, the pool of susceptible individuals depletes, leading to a deceleration in growth that can be modeled with logarithmic, logistic, or SIR models.
Logarithms in chemical kinetics are used to linearize exponential data, allowing for the determination of rate constants, activation energies, and reaction orders.
How to set up and solve exponential and logarithmic functions and equations:
Compound interest and continuous growth:
Model: A=Pe^rt for continuous compounding
Solve for time when you want to reach amount A: take natural log to get t=ln(A/P)/r
Use: planning investments or loans.
Decay and half‑life:
Model: N(t)=N0e^−kt.
Find half‑life: set N(t)=1/2N0 and solve t1/2=ln2/k.
Use: radiocarbon dating, pharmacokinetics.
Inverse problems with logarithms:
Question type: “How long until X reaches Y?”
Method: take ln of both sides of an exponential model and solve for the time variable.
Use: time to double an investment, time for cooling to a target temperature.
Example 1: Compound interest
Problem: $2,000 invested at 4% compounded continuously. How long to reach $3,000? Solution: 3000=2000e^0.04t. Take ln: t=ln(1.5)/0.04≈0.4055/0.04=10.14 years.
Example 2: Radioactive decay half‑life
Problem: A sample decays to 30% of its original amount in 5 years. Find half‑life. Solution: 0.30=e^−5k so k=−ln0.30/5. Then t1/2=ln2/k. Compute numerically for the half‑life.
Tips for modeling and interpretation: Choose the right model: exponential for unconstrained proportional change, logistic when there’s a limiting capacity. Check units: ensure the rate r or k matches the time units of t. Use logarithms to linearize: plotting ln(y) vs t turns y=Ae^kt into a straight line with slope k, useful for parameter estimation.