The exponential and the logarithm, their domains, ranges and graphs

What to assume they know

• Why what a function approaches need not be what it equals — domain, range, and reading a graph
• Index laws for real exponents, and that a positive base raised to any real power is positive
• The meaning of an inverse relationship between two functions, and reflection of a graph in the line y = x
• A relation is nothing more than a chosen part of the product — a function as a rule together with its domain and range
• Reading a curve that approaches an axis without meeting it

What they should be able to do

• State the domain and the range of the exponential function with base e
• State the defining equivalence for the logarithm to base e, and read it in both directions
• Deduce the logarithm's domain and range from the exponential's range and domain
• Sketch both graphs, marking the axis each approaches and the point at which each crosses an axis
• Explain why the two graphs are reflections of one another in the line y = x
• State what the section says about the size of e, and what it does not say
• Identify, from a printed graph, which of the two functions it shows
• Explain why the exponential never takes a value at or below zero, and what that means for the logarithm's domain

Where it usually goes wrong

• "e is about 2.718, so that is what the section says." It says e lies between 2 and 3, and nothing more precise. Anything sharper is imported from elsewhere, and an explanation should be honest about that, because the next topic's inequality quietly depends on e − 2 being a small positive number.
• "The logarithm is defined by a formula." It is defined by an equivalence with the exponential. There is no formula in this section, and everything about its domain and range is read off that equivalence.
• "The logarithm of 0 is 0" or "is very small." It is nothing at all: 0 is outside the domain, because no power of e equals 0.
• "The exponential's graph touches the horizontal axis on the left." It approaches without meeting, which is the drawn behaviour on p. 359 and the reason the range is strictly positive.
• "The two graphs are unrelated pictures the section happens to print together." They are reflections in y = x, and the interchange of domain and range is the algebraic shadow of that.
• "Because the section is in a supplement, it is optional." It carries a numbered section of Chapter 12 and its own exercise, and the spine indexes it. It is chapter content printed in an odd place.

Questions to check understanding

• State the domain and range of the exponential and of the logarithm to base e
• Convert between the exponential form and the logarithmic form of the same statement
• Sketch either graph and mark its intercept and its asymptotic behaviour
• Explain why the logarithm is not defined at 0 or at negative numbers
• One-mark: between which two whole numbers does e lie, as this section states it
• Given a graph, identify which of the two functions it shows and justify the answer from the domain

Examples worth working on the board

Values marked verified are an added derivation from what the section states; no answer key was consulted.

• Where the section is (Supplementary Material, printed pp. 357–364; this chapter's part begins at p. 359 under a heading naming Chapter 12). §12.6 is not printed in the chapter file at pp. 217–256, which stops at §12.5. Anyone working only from the chapter will not find this material at all.
• The number e (p. 359). Attributed to Leonhard Euler, dated 1707–1783, and described only as lying between 2 and 3. No decimal expansion and no defining series or limit for e appears on pp. 359 or 360. Whatever a student knows about e from elsewhere, this section supplies only the bracket.
• The exponential function (p. 359). Written as e raised to the power x, for x any real number. Its domain is stated to be the whole real line, and its range the set of positive reals.
• Fig. 13.11 (p. 359), read off the page image. A single curve on a pair of axes marked Y and X with the origin labelled O. To the left of the origin the curve runs close above the horizontal axis without meeting it; to the right it climbs steeply. The lettering beneath the curve names it as the graph of the exponential. The caption is printed as Fig. 13.11 — a chapter-13 number inside a section headed Chapter 12. Cite the printed string and say it is a leftover.
• The logarithmic function (p. 359). Written with base e, taking the positive reals to the reals. Defined by the equivalence that log of x equals y exactly when e raised to the power y equals x. Domain the positive reals, range the whole real line.
• Fig. 13.12 (p. 360), read off the page image. A single curve lying entirely to the right of the vertical axis, rising from far below near the axis, crossing the horizontal axis, and flattening as it goes right. Labelled beneath as the graph of the logarithm. Caption printed as Fig. 13.12, the same leftover.
• The two crossings (section 7; an added derivation from the definitions, not printed). Verified: e raised to the power 0 is 1, so the exponential's graph meets the vertical axis at height 1. Reading the same fact through the equivalence, the logarithm of 1 is 0, so the logarithm's graph meets the horizontal axis at 1. These two crossings are the same statement seen twice, and they are the anchor points to mark on both sketches. Both figures are drawn without numerical scales, so the crossings are not labelled on the printed page.
• The reflection (section 8; an added derivation, not stated in the section). Verified: the defining equivalence says the pair (y, x) lies on the exponential's graph exactly when the pair (x, y) lies on the logarithm's. Swapping the two coordinates of every point is reflection in the line y = x, so the graphs are mirror images in that line. This is why the two domains and ranges are interchanged rather than merely different.
• Why nothing at or below zero (section 9). Verified: the exponential's range is given as the positive reals, so the equivalence can only ever be satisfied when x is positive — which is exactly why the logarithm's domain excludes 0 and everything below it. The exclusion is a consequence, not an extra rule.

Figures to have open

• Both curves redrawn on one pair of axes with the line y = x, the two crossings marked at 1 on each axis, and the two asymptotic behaviours drawn accurately. The section prints the two graphs separately and without the line y = x; the combined picture is added here, and section 8's argument needs it.
• Fig. 13.11 redrawn alone (p. 359) for section 4, and Fig. 13.12 alone (p. 360) for section 7. Both printed without numerical scales; keep them unscaled so they match the page, and put the numbers only on the combined figure.
• A domain-and-range card: two number lines per function, inputs above and outputs below, drawn so the interchange is visible at a glance.
• No photograph is needed.

Where this sits in the book

• NCERT Class XI Mathematics, Supplementary Material, printed pp. 357–364. The Chapter 12 part begins on p. 359 under a heading giving the chapter number, with §12.6 "Limits Involving Exponential and Logarithmic Functions" running pp. 359–362. The introduction of both functions, with domains, ranges and the two graphs, occupies pp. 359–360.
• Fig. 13.11 (p. 359) and Fig. 13.12 (p. 360).
• The chapter file kemh112, printed pp. 217–256, does not contain §12.6. Chapter 12 as printed there ends at §12.5, followed by the Miscellaneous Examples, the Miscellaneous Exercise, the Summary and the Historical Note.
• The limits this section is building toward are in Two standard limits, the first squeezed out of an inequality and the second reduced to it.