ECON5005 · Quantitative Tools for Economics
Exponentials & Logarithms
Exponentials and logarithms are the growth-and-decay toolkit of ECON5005 Quantitative Tools for Economics at the University of Sydney. This chapter builds the index and log laws, change of base, and the single move behind every compounding, discounting and growth question — isolate the power, then take a log. It is the algebra that powers the financial-mathematics block and returns in the stability of difference equations later in the unit.
What this chapter covers
- 01Read an exponential function y = a^x: why every curve passes through (0, 1), and how the base sets growth vs decay
- 02Use the index laws fluently (product, quotient, power, negative and fractional exponents)
- 03Define a logarithm as the inverse of an exponential: a^x = b if and only if x = log_a(b)
- 04Apply the five log laws — product, quotient, power, the identities, and the exp/log inverse pair
- 05Convert any logarithm with change of base: log_a(b) = ln(b)/ln(a)
- 06Work with the natural base e and the natural logarithm ln in continuous-growth models
- 07Solve exponential equations a^x = k by taking logs so the exponent comes down
- 08Solve logarithmic equations by condensing to one log, switching to exponential form, and checking the domain
- 09Answer 'how long until…?' compounding and doubling-time questions with the time formula
- 10Avoid the classic traps: no law for log of a sum, and reject roots that make a log argument non-positive
How many years until output reaches a target?
- +1Set up and isolate the power: 600(1.04)^t = 900, so dividing by 600 gives (1.04)^t = 1.5.
- +1Take natural logs of both sides; the power law brings the exponent down: t · ln(1.04) = ln(1.5).
- +1Solve: t = ln(1.5) / ln(1.04) = 0.40546 / 0.039221 ≈ 10.34 years.
- +1Interpret: growth is credited yearly, so output first exceeds $900bn at the end of year 11 (after 10 years it is 600·1.04^10 ≈ $888bn; after 11 years ≈ $923bn).
Key terms
- Exponential function
- A function y = a^x with a fixed base a > 0, a ≠ 1, and the variable in the exponent. It passes through (0, 1) because a^0 = 1; a > 1 gives growth, 0 < a < 1 gives decay.
- Logarithm
- The inverse of an exponential: log_a(b) is the power you raise base a to in order to get b. Formally, a^x = b if and only if x = log_a(b).
- Natural logarithm (ln)
- The logarithm with base e, written ln x = log_e(x). It is the standard log in continuous-growth and calculus work.
- Euler's number e
- The constant e ≈ 2.71828, the base of continuous growth and of the natural logarithm. Continuous compounding uses S = P e^{rt}.
- Change of base
- Rewriting a logarithm in a base you can compute: log_a(b) = ln(b)/ln(a). This lets you evaluate or solve any a^x = k with a calculator that only has ln and log.
- Exponential growth / decay
- Change by a constant factor each period. A factor above 1 (e.g. 1.05) grows the quantity; a factor below 1 (e.g. 0.95) decays it toward zero.
- Compound interest
- Interest earned on both principal and accumulated interest: S = P(1 + r/m)^{mt}, where r is the nominal annual rate, m the compounds per year and t the years.
- Doubling / target time
- The number of periods for a quantity to reach a target, found by solving the exponential equation for the exponent: t = ln(S/P) / [m · ln(1 + r/m)].
Exponentials & Logarithms FAQ
Is there a law for the logarithm of a sum, like log(m + n)?
No. The log laws cover products, quotients and powers only: log_a(mn) = log_a(m) + log_a(n), log_a(m/n) = log_a(m) − log_a(n), and log_a(m^k) = k·log_a(m). There is no rule that simplifies log(m + n) — writing log(m + n) = log(m) + log(n) is one of the most common and most heavily penalised errors on this topic.
Why do I sometimes throw away an answer when solving a log equation?
A logarithm is only defined for a positive argument. When you condense a log equation, switch to exponential form and solve the resulting quadratic, you can get roots that make one of the original log arguments zero or negative. Those roots are not valid solutions. Always state the domain first and reject any root that breaks it — the rejection usually carries its own mark.
Can AI help me with exponentials and logarithms in ECON5005?
Yes, as a study aid. An AI tutor like Sia can explain the log laws step by step, walk you through change of base, and check your reasoning on a practice problem so you understand the method — for example, why you take logs to bring an exponent down. Use it to learn the technique on your own practice questions; it will not, and should not, hand you answers to graded quizzes, the mid-semester exam or the final, and no tool can guarantee a mark or a grade. The exams are in-person written papers, so the real goal is being able to do the working yourself.
Studying with AI? Sia — free AI economics tutor works through ECON5005 step by step.
Exam move
Treat this chapter as pure fluency: the log laws and change of base should be automatic before exam day, because they reappear inside the financial-mathematics and difference-equation questions rather than as an isolated topic. Drill the two templates until they are reflexes — to solve an exponential equation, isolate the power and take a log so the exponent comes down; to solve a log equation, condense to a single log, switch to exponential form, solve, then check the domain and reject invalid roots. Budget roughly two minutes per mark, keep four to five significant figures in any ln value until the final line, and always show your method, since these parts are marked step by step and clear working banks marks even when a number slips. Confirm the exact assessment weights and the calculator or formula-sheet policy on your Canvas site.
