Difficulty: Introductory–Intermediate | Prerequisites: Basic algebra, familiarity with exponential functions (Section 1.4 or equivalent)
This topic sits right at the bridge between algebra review and the calculus toolkit. Logarithms will reappear when you differentiate and integrate exponential functions later in the course, so a solid grip on what they are and how their rules work saves a lot of pain down the line. If you are comfortable with exponent rules and can graph y = b^x, you are ready for this. If not, revisit exponential functions first.
Every operation has an opposite that undoes it: subtraction undoes addition, division undoes multiplication. The same idea extends to functions: the logarithm is the function that undoes an exponential. Three tidy rules (product, power, and quotient) let you break apart or combine logarithmic expressions, which is how you isolate a variable trapped inside an exponent.
Inverse operation
An operation that reverses the effect of another operation, returning you to the original value. Addition and subtraction are inverse operations; so are multiplication and division.
In simple terms: it is the "undo button" for a mathematical step.
Inverse function
A function that reverses the input-output relationship of another function. If f takes x to y, then f's inverse takes y back to x.
Think of it as: swapping the input and the output. If you know the output, the inverse tells you what went in.
Logarithm (log)
The inverse function of an exponential. log_b(x) answers the question: "To what power must I raise b to get x?"
In simple terms: if b^y = x, then log_b(x) = y. The log "undoes" the exponential.
Natural logarithm (ln)
The logarithm with base e (approximately 2.718). Written ln(x), it is shorthand for log_e(x). It is the inverse of e^x.
Think of it as: the special log that pairs with the most common exponential in calculus, e^x.
Base of a logarithm
The number b in log_b(x). Each exponential b^x has its own matching log with the same base. For instance, log_2 is the inverse of 2^x, and log_45 is the inverse of 45^x.
In simple terms: the base tells you which exponential you are "undoing."
Subtraction is the inverse of addition, and vice versa: x + 8 - 8 = x. Adding 8 does something to x; subtracting 8 undoes it.
Division is the inverse of multiplication: (x · 7) / 7 = x. Multiplying by 7 does something to x; dividing by 7 undoes it.
The pattern: apply an operation, then apply its inverse, and you are back where you started.
The same "undo" idea scales up to entire functions. If a function f maps x to some output, the inverse function f⁻¹ maps that output back to x.
The exponential function b^x has a specific inverse: log_b. Applying log_b to b^x returns x.
log_b(b^x) = x
Every base has its own pair: log_2 inverts 2^x, log_45 inverts 45^x, log_8.719 inverts 8.719^x.
The most important special case in calculus: the inverse of e^x is log_e(x), written ln(x), the natural logarithm.
Logarithms are functions in their own right, not just a notational convenience. They pass the vertical line test (each input has exactly one output).
Before calculators existed, people relied on printed tables of log values to perform multiplication and division quickly.
The shape of the log curve: it rises steeply at first, then levels off. If you plot "time spent cleaning" on the x-axis and "happiness" on the y-axis, the log shape captures diminishing returns nicely: early effort yields a big payoff, but the gains slow down over time.
These rules hold for any base b (provided a > 0 and b > 0).
Product rule (Rule 1): log(ab) = log(a) + log(b). A product inside the log splits into a sum of logs.
Power rule (Rule 2): log(a^b) = b · log(a). An exponent inside the log comes out front as a multiplier.
Quotient rule (Rule 3): log(a / b) = log(a) - log(b). A fraction inside the log splits into a difference of logs.
These three rules are the main tools you will use whenever you need to rearrange or simplify expressions containing logs.
\log(ab) = \log(a) + \log(b)\log(a^{b}) = b \cdot \log(a)\log\!\left(\frac{a}{b}\right) = \log(a) - \log(b)\log_b(x) = \frac{\ln(x)}{\ln(b)}The last formula is the change-of-base rule. It lets you convert any logarithm into a ratio of natural logs, which is what your calculator or computer uses internally.
Logarithmic scales appear everywhere data spans many orders of magnitude. The Richter scale for earthquakes is logarithmic: a magnitude-7 quake releases roughly 31.6 times the energy of a magnitude-6 quake. The decibel scale for sound works the same way. In finance and biology, logarithms turn exponential growth curves into straight lines, making trends easier to read and compare.
Students often think log(a + b) = log(a) + log(b). It does not. The product rule applies to log(a · b), not to log(a + b). There is no shortcut for the log of a sum.
Students sometimes treat "ln" and "log" as the same function. On most calculators, "log" means log base 10, while "ln" means log base e. In pure mathematics, "log" with no base sometimes defaults to ln, which adds to the confusion. Always check which convention your course uses.
The notation f⁻¹(x) looks like an exponent, and students occasionally read it as 1/f(x). The superscript -1 here means "inverse function," not "reciprocal."
Students sometimes apply the power rule in the wrong direction, writing log(a^b) = a · log(b) instead of b · log(a). The exponent is what comes out front, not the base.
⚠️ Expect to be asked to simplify or expand logarithmic expressions using the three log laws. These appear on nearly every exam covering this section.
⚠️ The change-of-base formula (log_b(x) = ln(x) / ln(b)) is frequently tested, especially in a "solve for x" context where the variable sits in the exponent.
⚠️ Know the relationship between an exponential and its inverse log cold: log_b(b^x) = x and b^(log_b(x)) = x. You will need this to verify solutions.
⚠️ Be ready to distinguish between the natural log ln and log base 10. Problems may specify one or the other, and mixing them up changes the answer.
True or false: log(a + b) = log(a) + log(b).
Fill in the blank: log_b(b^x) = ______.
True or false: ln(x) is another way of writing log_10(x).
Fill in the blank: log(a^5) = ______ · log(a).
True or false: the change-of-base formula lets you rewrite log_7(x) as ln(x) / ln(7).
Answers: 1. False. 2. x. 3. False (ln is log base e). 4. 5. 5. True.
Q: Use the log laws to expand ln(5x³).
A: ln(5x³) = ln(5) + ln(x³) = ln(5) + 3 ln(x), using the product rule first and then the power rule.
Q: Simplify log(12) - log(3) into a single logarithm.
A: log(12) - log(3) = log(12 / 3) = log(4), using the quotient rule.
Q: Rewrite log_5(x) using only the natural logarithm.
A: log_5(x) = ln(x) / ln(5), by the change-of-base formula.
Q: If log_b(b^7) = k, what is k?
A: k = 7. The log base b and b^x are inverse functions, so log_b(b^7) = 7.
Q: A student writes log(a²) = a · log(2). What mistake did they make?
A: They swapped the base and the exponent. The power rule says log(a²) = 2 · log(a). The exponent comes out front, not the base.
This material connects directly to differentiation of exponential and logarithmic functions, which you will meet shortly. The derivative of ln(x) is 1/x, and the derivative of e^x is e^x, both of which rely on the inverse relationship covered here. The log laws also resurface during integration when you encounter integrals of the form 1/x dx = ln|x| + C.
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