4.6 Chapter Review
4.6.1 Chapter summary
Exponential models describe constant proportional change, while logarithms recover exponents and invert exponential functions. Logarithm laws support simplification and equation solving, but require positive arguments. Logarithmic scales reveal multiplicative structure, and flow-duration curves summarize how often streamflows meet or exceed selected values.
4.6.2 Common mistakes
- Using the percent rate as the growth factor.
- Forgetting that logarithmic inputs must be positive.
- Splitting the logarithm of a sum.
- Failing to check solutions in the original logarithmic equation.
- Confusing an exceedance percentage with the percentage of observations below a flow.
4.6.3 Exercises
- Find the value after 7 periods for \(A(t)=50(1.08)^t\).
Check Your Work
Approximately 85.69. - Find the domain of \(\ln(4-3x)\).
Check Your Work
\(x<4/3\). - Expand \(\log(x^5/\sqrt y)\).
Check Your Work
\(5\log x-(1/2)\log y\). - Solve \(3^x=20\).
Check Your Work
\(x=\ln20/\ln3\approx2.727\). - Solve \(\ln x+\ln(x-1)=\ln6\).
Check Your Work
\(x=3\). - Interpret a discharge of 16 m³/s at 75% exceedance.
Check Your Work
Discharge equals or exceeds 16 m³/s about 75% of the time.