Logarithmic Functions

Part 14, Chapter 14: Functions and Their Graphs

Learning objectives

Define the logarithm as the inverse of the exponential
Evaluate logarithms of perfect powers
Apply the three laws of logarithms
Solve equations involving logarithms

A logarithm answers the question "what exponent?" Multiplication is repeated addition; exponentiation is repeated multiplication; and the logarithm undoes that repeated multiplication by counting how many times you had to multiply. Once you internalise that, the three laws of logarithms read like obvious arithmetic facts about exponents in disguise.

Definition by inversion

For $b > 0$ , $b \neq 1$ , the logarithm base $b$ of $x$ is the unique number $y$ satisfying $b^y = x$ . We write $y = \log_b x$ bx. Equivalently:

$y = \log_b x \iff b^y = x.$ bxiffby=x.

Examples: $\log_2 8 = 3$ because $2^3 = 8$ ; $\log_{10} 1000 = 3$ 101000=3 because $10^3 = 1000$ ; $\log_5 1 = 0$ because $5^0 = 1$ .

The three laws

For positive $x$ and $y$ :

Product rule: $\log_b(xy) = \log_b x + \log_b y$ b(xy)=logbx+logby.

Quotient rule: $\log_b(x/y) = \log_b x - \log_b y$ b(x/y)=logbx−logby.

Power rule: $\log_b(x^n) = n \log_b x$ .

Why do these hold? Write $x = b^{\alpha}$ and $y = b^{\beta}$ ; then $xy = b^{\alpha + \beta}$ , so its logarithm is $\alpha + \beta = \log_b x + \log_b y$ . Each rule is just a restatement of an exponent rule, viewed through the logarithm lens.

Two special bases

The natural logarithm uses base $e$ : $\ln x = \log_e x$ ex. The common logarithm uses base $10$ : in many engineering contexts $\log x$ means $\log_{10} x$ 10x by convention. (Pure-math texts often use $\log x$ for $\ln x$ instead. Check the convention of the book you are reading.)

Change of base

To convert from base $b$ to base $c$ :

$\log_b x = \dfrac{\log_c x}{\log_c b}.$ bx=dfraclogcxlogcb.

This is how a calculator that only knows $\ln$ and $\log_{10}$ 10 can compute $\log_2 13$ : as $\dfrac{\ln 13}{\ln 2}$ .

Where this shows up

Information Theory: The Shannon entropy $H = -\sum p_i \log_2 p_i$ ilog2pi measures information content in bits; the logarithm here is the same one you are learning, and it shows up in every compression algorithm (gzip, zstd, mp3).

Chemistry: The pH scale is $-\log_{10}[H^+]$ ; pH 4 is ten times more acidic than pH 5 because of the logarithm. Same trick on the Richter scale (earthquakes), decibel scale (sound), and apparent magnitude (stars).

Algorithms: Binary search is $O(\log n)$ because doubling the input adds one step; the same logarithm explains why a sorted array of one trillion items can be searched in under 40 comparisons.

(Switch the widget to Product rule mode and slide $a$ and $c$ . The widget draws $\log_b(a)$ b(a) and $\log_b(c)$ b(c) as separate heights on the logarithm curve, then shows that their sum equals $\log_b(ac)$ b(ac). The product rule is the law made geometric.)

Solving logarithmic equations

To solve $\log_5 x = 3$ , convert to exponential form: $x = 5^3 = 125$ . To solve $\log_2(x + 1) = 5$ , the same trick gives $x + 1 = 2^5 = 32$ , so $x = 31$ . The general technique: exponential-and-logarithm undo each other, so to remove a logarithm, exponentiate; to remove an exponential, take a logarithm.

Try it

Predict first: by the product rule, $\log_{10}(6)$ 10(6) equals $\log_{10}(2) + \log_{10}(?)$ ? Switch to Product rule mode, set $a = 2, c = 3, b = 10$ , and confirm by hand that $\log_{10}(6) = \log_{10}(2) + \log_{10}(3)$ 10(6)=log10(2)+log10(3).

Switch the base to $b = e$ (slide $b$ to about $2.72$ ). Pick a value $x$ on the log curve and verify $e^{\ln x} = x$ .

Compute $\log_9 27$ by hand using the change-of-base formula in two different ways (base 3, then natural log), you should get $3/2$ .

Pause: why is $\log_b 1 = 0$ b1=0 for every base? Why is $\log_b b = 1$ bb=1? Tie both to the exponential definition.

Try it in code

A trap to watch for

The single most common error in pre-calculus: writing $\log(a + b) = \log a + \log b$ . This is wrong. The product rule says $\log(a \cdot b) = \log a + \log b$ , the inside is a product, not a sum. Test: $\log_{10}(1 + 1) = \log_{10}(2) \approx 0.301$ approx0.301, while $\log_{10}(1) + \log_{10}(1) = 0 + 0 = 0$ 0+0=0. These are not equal. There is no nice identity for the logarithm of a sum; if a problem hands you $\log(a + b)$ , leave it. The trap exists because $\log$ looks like a linear operator (it is written before an argument, like a function), but its action on multiplication, not addition, is what produces the nice identity.

What you now know

You can convert between logarithmic and exponential form, apply the three laws to simplify expressions, change base when needed, and solve simple logarithmic equations by exponentiating. The next chapter generalises functions to mappings between abstract sets, the same definition you learnt in 13.1, but stripped of any reliance on numbers, ready for the rest of mathematics.

Quick check

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References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 13 §5, logarithms defined as inverses of exponentials.

Stewart, J. (2015). Calculus: Early Transcendentals, 8th ed. Cengage. Section 6.3: the three laws and change of base.

Apostol, T. M. (1967). Calculus, Volume I, 2nd ed. Wiley. Chapter 6: rigorous construction of $\ln x$ as an integral and the resulting identities.

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