Difficulty: Intermediate | Prerequisites: Exponent rules, basic algebra
This chapter introduces logarithmic functions as the inverse of exponential functions, a relationship that underpins most of the calculus you will do with exponentials later in the course. If you are comfortable with exponent rules, you are ready for this. If not, review those first, because every logarithm property is an exponent rule turned inside out. The natural logarithm (ln) gets special attention here because it is the version that appears in continuous compounding, derivative proofs, and nearly every applied calculus formula from this point forward.
Logarithms reverse what exponents do: if b raised to a power gives you some number, the logarithm tells you what that power was. There is a handful of clean rules (product, quotient, power) that let you break apart or combine log expressions, and these come up constantly in simplification and solving. The natural logarithm, ln, is simply log base e, and it is the one you will use most from here on because e is the base that makes calculus with exponentials work.
Logarithm (log)
The exponent you must raise a given base to in order to get a particular number. Formally: log_b(x) = y means b^y = x.
Think of it as the "undo" button for exponentiation. If 2^5 = 32, then log_2(32) = 5.
Logarithmic base
The number being raised to a power. In log_b(x), the base is b. The base must be positive and cannot equal 1.
In simple terms, this is the "bottom number" that does the repeated multiplying.
Common logarithm (log_10 or just "log")
A logarithm with base 10. Often written without the base when context is clear.
Think of it as the log you use for orders of magnitude: log(1000) = 3 because 10^3 = 1000.
Natural logarithm (ln)
A logarithm with base e (approximately 2.718). Written as ln(x), which is shorthand for log_e(x).
In simple terms, ln is the special log that pairs with the number e. It shows up everywhere in calculus because e is the base that makes derivatives of exponentials clean.
Inverse function relationship
The exponential function e^x and the natural log ln(x) undo each other. That is: e^(ln(x)) = x and ln(e^x) = x.
Think of it as two machines in a row where the second machine always reverses what the first one did.
For any valid base b: log_b(1) = 0, because b^0 = 1
For any valid base b: log_b(b) = 1, because b^1 = b
If log_b(u) = log_b(v), then u = v (one-to-one property)
These three rules mirror the exponent rules for multiplication, division, and powers.
Product rule: log_b(uv) = log_b(u) + log_b(v). Multiplying inside the log becomes addition outside.
Quotient rule: log_b(u/v) = log_b(u) - log_b(v). Dividing inside becomes subtraction outside.
Power rule: log_b(u^r) = r * log_b(u), for any real number r. An exponent inside pulls out front as a multiplier.
One more identity to keep handy: log_b(b^n) = n. The log and the base cancel, leaving just the exponent.
Expanding means using the product, quotient, and power rules to break a single log into several simpler terms.
Example: log_3(x^3 * y^(-4))
= log_3(x^3) + log_3(y^(-4)) (product rule)
= 3*log_3(x) + (-4)*log_3(y) (power rule)
= 3log_3(x) - 4log_3(y)
Example: log_2(y^5 / x^2)
= log_2(y^5) - log_2(x^2) (quotient rule)
= 5log_2(y) - 2log_2(x) (power rule)
Example: log_7(x^3 * sqrt(1 - y^2))
= log_7(x^3) + log_7((1 - y^2)^(1/2)) (product rule, rewriting the root)
= 3*log_7(x) + (1/2)*log_7(1 - y^2) (power rule)
All of the properties above apply when the base is e. Additionally:
ln(e) = 1
ln(1) = 0
ln(e^x) = x (the log undoes the exponential)
e^(ln(x)) = x (the exponential undoes the log)
ln(2) does not simplify to a clean number. It is approximately 0.693.
The relationship between e^x and ln(x) is that they are inverse functions. Graphically, one is the reflection of the other across the line y = x.
A common pattern: the square root of x equals x^(1/2), so log_e(sqrt(x)) = (1/2)*ln(x). This conversion from root notation to fractional exponents is a necessary first step before applying the power rule.
log_10(1000) = 3, because 10^3 = 1000
log_2(32) = 5, because 2^5 = 32
log_5(1/25) = -2, because 5^(-2) = 1/25
Rule | Formula | What it does |
|---|---|---|
Identity (1) | log_b(1) = 0 | Log of 1 is always 0 |
Identity (base) | log_b(b) = 1 | Log of the base is always 1 |
Product rule | log_b(uv) = log_b(u) + log_b(v) | Multiplication becomes addition |
Quotient rule | log_b(u/v) = log_b(u) - log_b(v) | Division becomes subtraction |
Power rule | log_b(u^r) = r * log_b(u) | Exponent pulls out front |
Inverse (log then exp) | b^(log_b(x)) = x | Exponential undoes the log |
Inverse (exp then log) | log_b(b^x) = x | Log undoes the exponential |
Natural log inverse | e^(ln(x)) = x and ln(e^x) = x | e and ln cancel each other |
Logarithms are how you solve for time in compound interest problems: "how long until my investment doubles?" turns into an equation you solve with ln. This is exactly the application covered later in this chapter.
Beyond finance, logarithmic scales appear wherever quantities span huge ranges: the Richter scale for earthquakes, the decibel scale for sound, and pH for acidity are all logarithmic. In business analytics, log transformations are used to make skewed data (like income distributions or website traffic) easier to model and visualise.
Students often think log_b(u + v) = log_b(u) + log_b(v). It does not. The product rule applies to log_b(u * v), not log_b(u + v). There is no rule for the log of a sum.
Students often think ln(2) simplifies to a clean fraction or integer. It does not. ln(2) is irrational, approximately 0.693. Leave it as ln(2) unless told otherwise.
Students sometimes confuse the base and the argument. In log_5(25), the 5 is the base and 25 is the argument. The question is: "5 raised to what power gives 25?" Swapping these gives a completely different (and wrong) answer.
Students sometimes try to "cancel" a log by dividing. You undo a log by exponentiating (raising the base to both sides), not by dividing by log.
⚠️ Expanding and condensing log expressions is a near-certain exam topic. Know the product, quotient, and power rules cold.
⚠️ Expect at least one problem that asks you to solve for time using ln, typically in a compound interest context.
⚠️ The inverse relationship e^(ln(x)) = x and ln(e^x) = x comes up in proofs and in simplifying derivatives. If you cannot apply it automatically, you will get stuck on later material.
⚠️ Negative and fractional exponents inside logs (like log_5(1/25) or log of a square root) are common exam tricks. Convert roots to fractional exponents and reciprocals to negative exponents before applying the rules.
True or false: log_b(u + v) = log_b(u) + log_b(v). Answer: False. The product rule requires multiplication inside the log, not addition.
Fill in the blank: ln(e^7) = ___. Answer: 7.
True or false: log_2(1) = 1. Answer: False. log_2(1) = 0, because 2^0 = 1.
Fill in the blank: e^(ln(5)) = ___. Answer: 5.
True or false: log_3(9) = 3. Answer: False. log_3(9) = 2, because 3^2 = 9.
Q: Expand log_3(x^5 * y^(-2)) into separate terms.
A: 5log_3(x) - 2log_3(y). Apply the product rule first, then the power rule to each term.
Q: Simplify ln(e^(4x)).
A: 4x. The natural log and e are inverses, so they cancel.
Q: Evaluate log_2(1/8).
A: -3. Rewrite 1/8 as 2^(-3), then log_2(2^(-3)) = -3.
Q: Expand log_7(x^2 * sqrt(1 - y^2)) completely.
A: 2*log_7(x) + (1/2)*log_7(1 - y^2). Rewrite the square root as a 1/2 power, apply the product rule, then the power rule.
Q: If log_b(u) = log_b(v), what can you conclude?
A: u = v. This is the one-to-one property of logarithms. It is the basis for solving logarithmic equations by setting the arguments equal.
This material connects directly to continuous compound interest (covered in Part 2 of these notes), where solving for time requires taking the natural log of both sides. Without fluency in log properties, you will stall on those problems.
Logarithmic differentiation, which appears in later calculus chapters, depends on the same product and quotient rules used here, applied to ln. If you can expand a log expression now, the differentiation step later is largely mechanical.
The inverse relationship between e^x and ln(x) is also foundational for integration techniques (substitution and integration by parts) in Calculus II.
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