Teaching Notes · Math for Social Sciences

Exponential and Logarithmic Functions in Finance

Every earlier function in this course, lines, polynomials, rational functions, changes by a fixed amount or a fixed rule applied to the input. Money under interest does not behave that way: it grows by a percentage of whatever it has already become, which is exactly what an exponential function describes. This set of notes builds the exponential function and its inverse, the logarithm, then puts them straight to work on two of the most common questions in finance: what a deposit becomes over time, and how much has to be deposited now to reach some target later.

The exponential function and the constant e

An exponential function has the form f(x) = aˣ, where the base a is positive and not equal to 1. Exponential functions share a small set of properties no matter what the base is: the domain is every real number, the range is every positive number, the function grows without bound as x increases, and it shrinks toward (but never reaches) zero as x decreases. Among all possible bases, one shows up again and again in calculus: the irrational number e, approximately 2.71828, defined formally as the limit of (1 + x)^1/x as x approaches 0, or equivalently of (1 + 1/n)ⁿ as n grows without bound. The function f(x) = eˣ, the natural exponential function, behaves exactly like any other exponential function, just with this particular base.

Compound interest: discrete and continuous

If P dollars are deposited at an annual interest rate r, compounded n times per year, the balance after t years is A = P(1 + r/n)^nt. As n is allowed to grow larger and larger, interest is credited more and more often, and in the limit as n approaches infinity, the formula converges to A = Pe^rt, the balance under continuous compounding, where interest is credited constantly rather than at fixed intervals.

The difference between the two is small per period but adds up. Suppose 12,000 is deposited for 25 years at an annual rate of 7 percent. Compounded quarterly (n = 4), the balance is A = 12,000(1 + 0.07/4)¹⁰⁰ = 68,017.87. Compounded continuously, the balance is A = 12,000·e^0.07×25 = 69,055.23, over a thousand dollars more on the same deposit, same rate, same term, purely from compounding without pause rather than four times a year.

Present value: working backward from a future goal

Compound interest answers how much a deposit grows into. Present value asks the reverse question: given a target balance A wanted t years from now, at a known rate r compounded n times per year, how much P should be deposited today? Solving the compound interest formula for P gives P = A / (1 + r/n)^nt.

An investor wants a certificate of deposit to be worth 15,000 after 12 years, at an annual rate of 8 percent compounded monthly. Here A = 15,000, r = 0.08, n = 12, and t = 12, so the present value is P = 15,000 / (1 + 0.08/12)¹⁴⁴ = 5,761.72. Depositing that amount today, at that rate, over those 12 years, reaches the 15,000 target exactly, which is the entire logic behind pricing a bond or planning for a future liability: work the compound interest formula backward from the target instead of forward from the deposit.

Differentiating the exponential function

The natural exponential function has a property no other function in this course has: its own derivative is itself, d/dx(eˣ) = eˣ. Combined with the chain rule, this extends to any exponent that is itself a function of x: d/dx[e^u(x)] = e^u(x)·u'(x). For f(x) = e^2x, the derivative is f'(x) = 2e^2x. For f(x) = x²eˣ, built from a product, the derivative is f'(x) = eˣ(2x + x²), combining the product rule with this same exponential rule.

Logarithms as the inverse operation

A logarithm answers the question an exponential function asks in reverse: given a result, what exponent produced it? The logarithm of x to base a, written log_a x, equals y exactly when a^y = x. In calculus the base that matters most is again e, giving the natural logarithm, written ln x, defined so that ln x = y exactly when e^y = x. Because the natural logarithm and the natural exponential function undo each other, ln(eˣ) = x and e^{ln x} = x for every valid x, and in particular ln 1 = 0 and ln e = 1.

Three properties let a complicated logarithmic expression be rewritten, in either direction: ln(xy) = ln x + ln y, ln(x/y) = ln x − ln y, and ln(xⁿ) = n ln x. These make some expressions much easier to differentiate once broken apart. The expression ln[x²(x + 1)], for instance, rewrites as 2 ln x + ln(x + 1), and in the other direction, ln x + 2 ln y rewrites as ln(xy²), collapsing a sum of logarithms back into a single one.

Solving exponential and logarithmic equations

The same inverse relationship solves equations that would otherwise have the unknown trapped in an exponent or inside a logarithm. To solve 10 + e^0.1x = 14, isolate the exponential term first: e^0.1x = 4. Taking the natural logarithm of both sides and using ln(eˣ) = x gives 0.1x = ln 4, so x = 10 ln 4. To solve 3 + 2 ln x = 7, isolate the logarithm first: ln x = 2, then exponentiate both sides, e^{ln x} = e², giving x = e². In both cases the unknown only becomes solvable once the exponential or the logarithm has been undone by its inverse.

Differentiating the logarithmic function

The derivative of the natural logarithm is simple on its own, d/dx(ln x) = 1/x, and extends through the chain rule to d/dx[ln u(x)] = u'(x)/u(x) for any differentiable u. For f(x) = ln(2x² + 4), the derivative is f'(x) = 4x/(2x² + 4), which simplifies to 2x/(x² + 2). For f(x) = x² ln x, built again from a product, the derivative is f'(x) = 2x ln x + x, the ordinary product rule applied with 1/x standing in for the derivative of ln x.

Exponential growth and its inverse turn up everywhere in finance once you start looking: a bank balance under compound interest, the present value of a future obligation, and later in this course, a stream of saved deposits added up through a geometric series, all rest on the same pair of functions covered here.

Source: lecture notes for an introductory calculus course for the social sciences, covering exponential functions, the natural exponential function and the constant e, compound interest and present value, derivatives of exponential functions, logarithmic functions and their properties, solving exponential and logarithmic equations, and derivatives of logarithmic functions.