Exponential and logarithmic functions, and the two derivatives that make them worth having

What to assume they know

• Powers with whole-number exponents, and comparing them at a common input
• Indices: multiplying powers of a common base, and a power of a power
• The chain rule, and naming an inner and an outer function
• The quotient rule, and the product rule
• Reading a graph for its domain, its range, and the axes it approaches
• Reflection of a curve in the diagonal line, and what it does to a function
• The series whose sum names the base of natural logarithms, from Class XI
• The derivative of the inverse cosine, from the previous topic

What they should be able to do

• Compare the growth of two whole powers at a common input and say which is steeper
• Give one input at which a fixed exponential overtakes a named power, with the arithmetic
• State Definition 3 and list the five features of an exponential graph
• Name the base of the natural exponential function and say where it comes from
• State Definition 4 and convert between a power statement and a logarithm statement
• Give the domain and the range of the logarithm function and relate them to the exponential's
• Explain the reflection in Fig 5.11 and what it says about the two functions
• Prove the change-of-base rule and the product rule for logarithms from exponent arithmetic
• Say for which inputs the exponential of a logarithm returns the input
• Quote Theorem 5 and use both parts with the chain rule on the exercise set

Where it usually goes wrong

• "The section proved that the exponential outgrows every power." It checked one case at one input and said outright that it would not prove the general claim. Two of the section's own sentences say this.
• "Theorem 5 is proved in the chapter." It is stated and its proof is sent elsewhere by a footnote. Every derivative in the last four sections of the chapter rests on it.
• "An unsubscripted logarithm means base ten." In this chapter it means the natural base, fixed in italics on Part I p. 127. Reading it as base ten breaks the derivative in Theorem 5.
• "The logarithm of a negative number is negative." There is no such thing. The domain is the positive numbers, and Example 25 exists to make the point.
• "The exponential and the logarithm cancel each other in both orders for every input." Only where each is defined. Example 25 restricts the cancellation to positive inputs.
• "The curve meets the axis it approaches." Both graphs approach an axis and neither reaches it — the chapter says so twice, once in each list of features, in the same parenthesis.
• "The exponential and the logarithm are different kinds of object." They are reflections of each other in the diagonal, which is Fig 5.11's whole content; the domain of each is the range of the other.
• "The change-of-base rule needs calculus." It is five lines of exponent arithmetic and the chapter proves it that way, before any derivative appears.

Questions to check understanding

• Convert between a power statement and a logarithm statement, at a named base
• Give the domain and range of the exponential and of the logarithm, and say how the two pairs are related
• Prove the change-of-base rule from exponent arithmetic
• Prove that the logarithm of a product is the sum of the logarithms, and deduce the whole-power form
• Say for which inputs the exponential of a logarithm returns the input, and justify the restriction — the form of Example 25
• Differentiate a composite built from an exponential or a logarithm — the form of Exercise 5.4 Q1 to Q10
• Differentiate a logarithm at a base other than the natural one, using the change-of-base rule first — the form of Miscellaneous Example 39, part two

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The opening warning of §5.4 (Part I p. 125). The section announces that many of its statements are motivational and that precise proofs are outside the book's scope. Show this at the start, because it is the sentence that makes sections 2 to 5 honest — they are illustration, and the chapter says so before it begins.
• Fig 5.9 (Part I p. 126). Read off a three-hundred-dot the printed page: five curves on one pair of axes, each labelled along its own length — the first power, the square, the cube, the fourth power and the exponential at base ten. The four power curves all pass through the point at one and one, which is marked with a filled dot; the exponential passes through the point at zero and one, also marked, and the two marked points are joined by a horizontal dashed segment. All five curves are drawn to the right of the vertical axis only, apart from the exponential's flat tail running in from the left along the horizontal axis. The chapter's own point is that a higher exponent buys a steeper climb.
• The comparison of two large powers (Part I p. 126). The chapter takes the tenth and fifteenth powers and lets the input run from one to two, so that one reaches the tenth power of two and the other the fifteenth. Verified: the second is thirty-two times the first, which is exactly the point being made, and neither number needs to be evaluated for the comparison to land.
• The overtaking calculation (Part I p. 126). The chapter compares the exponential at base ten against the hundredth power at the input one thousand. Verified: the hundredth power of a thousand is ten to the three hundred, while ten raised to a thousand is ten to the thousand. Both exponents should be shown as exponents, because writing either out is impossible and that is itself the lesson. The chapter then says it will not prove the general claim.
• Definition 3 and the five features (Part I pp. 126–127). The exponential function at a base above one, followed by five bullets: the domain is the whole line; the range is the positive numbers; the point at zero and one is always on the graph; the graph rises from left to right; and for large negative inputs the curve draws close to the horizontal axis without meeting it. Verified: the third bullet is the statement that any base above one raised to the zeroth power is one, which the chapter says outright.
• The natural base (Part I p. 127). The chapter recalls from a Class XI appendix that a certain series sums to a number between two and three, names it, and builds the natural exponential on it. Verified: the series is one plus the reciprocals of the factorials, and its sum lies between two and three. The reference is to Class XI, not to anything in this chapter.
• Definition 4 and the worked conversions (Part I p. 127). The logarithm of a number at a base is the exponent that the base must carry to reach it. The chapter then converts four statements. Verified: two cubed is eight, so the logarithm of eight at base two is three; ten to the fourth is ten thousand, so the logarithm of ten thousand at base ten is four; and six hundred and twenty-five is both the fourth power of five and the square of twenty-five, so its logarithm is four at base five and two at base twenty-five. That last pair is the best thirty seconds in the section — one number, two bases, two different logarithms, and the change-of-base rule visible in miniature.
• The logarithm as a function, and the notation convention (Part I p. 127). Positive inputs to all real outputs. The chapter then fixes, in italics, that an unsubscripted logarithm in this chapter means the natural one. This convention is used silently for the rest of the chapter and a student who misses it will read every later derivative at the wrong base.
• Fig 5.10 (Part I p. 127). Three logarithm curves at bases two, the natural base and ten, each labelled, all through the point at one and zero, which is marked. Read off the pack render at a hundred dots per inch, all three plunge towards the vertical axis near the origin without touching it.
• The six features of the logarithm graph and Fig 5.11 (Part I p. 128). Domain the positive numbers; range the whole line; the point at one and zero always on the graph; rising from left to right; and near zero the value drops below every real number, with the curve approaching the vertical axis. The sixth bullet points at the figure. Read off the pack render: the exponential and the natural logarithm drawn on one pair of axes with the diagonal line drawn dashed between them, and the two points at zero and one and at one and zero both marked. The exchange is the content: the first two features of each list are the other's, swapped.
• The change-of-base proof (Part I p. 128). Name the three logarithms, convert each into a power statement, substitute one into another, and compare exponents. Verified: the substitution turns one base into a power of the other and the comparison of exponents gives the quotient form directly. Five lines, no calculus, and it is one of only two things proved in the section.
• The product proof and its two consequences (Part I pp. 128–129). The same method: convert to powers, multiply, add exponents. Verified. The chapter then specialises to a repeated factor, giving twice the logarithm, and states the whole-power version as an exercise for the reader, followed by the quotient version, also handed over. It adds that the power version holds for any real exponent and that it will not prove that. A printed slip in these two displayed lines is recorded in Notes.
• Example 25 (Part I p. 129). Is the exponential of the logarithm of an input always the input? Verified: only for positive inputs, because the logarithm has no value at zero or below. The chapter's argument takes the logarithm of both sides and uses that the logarithm of the natural base is one. This is the section's best guard against a student cancelling the two operations by reflex.
• Theorem 5 and its footnote (Part I p. 129). Two derivatives: the natural exponential is its own derivative, and the natural logarithm differentiates to the reciprocal. The chapter says it is skipping the proof, and a footnote at the foot of the page points the reader to supplementary material on a printed page well beyond this chapter. That page is not part of Chapter 5 and is not in the evidence pack; see Notes. State both derivatives and say plainly that the chapter does not derive them.
• Example 26 (Part I pp. 129–130). Four chain-rule applications: the exponential of the negated input; the sine of a logarithm; the inverse cosine of an exponential; and the exponential of a cosine. Verified: the first gives the negative of itself; the second gives the cosine of the logarithm divided by the input; the third gives the negative of the exponential over the square root of one minus its square, which uses the previous topic's inverse cosine derivative; the fourth gives the negative of the sine times the whole exponential.
• Exercise 5.4 (Part I p. 130). Ten items. Verified, in order: Q1 needs the quotient rule and gives the exponential times the difference of sine and cosine, over the square of the sine. Q2 gives itself divided by the square root of one minus the square. Q3 gives three times the square times itself. Q4 needs three stages. Q5 gives the negative of the exponential times the tangent of the exponential. Q6 is a sum of five terms whose derivatives carry the factors one to five. Q7 rewrites as the exponential of half the square root, giving itself over four times the square root. Q8 gives the reciprocal of the input times its logarithm. Q9 needs the quotient rule. Q10 is a cosine of a sum, giving the negative of the sine of that sum times the reciprocal plus the exponential. Q6 and Q7 are the two worth doing — the first because the pattern is visible only when all five terms are written out, and the second because simplifying before differentiating turns four lines into one.
• Miscellaneous Example 39, part two (Part I p. 141). The logarithm at base seven of a logarithm. Verified: the change-of-base rule turns it into a quotient with a constant denominator, and two chain-rule steps then give the reciprocal of the product of the input, the constant logarithm and the inner logarithm. The item exists to show that the change-of-base rule of this section is a differentiation tool, not just an identity.

Figures to have open

• A redraw of Fig 5.9 (Part I p. 126) with all five curves labelled along their lengths, the two marked points, and the horizontal dashed segment joining them. Draw the curves only where the chapter draws them.
• A five-row table of the exponential graph's features for section 4, and the matching six-row list for the logarithm in section 7. Use the repo's DataTable component.
• A redraw of Fig 5.11 (Part I p. 128) with the diagonal drawn dashed, both curves labelled, and the two marked points joined by a construction line showing the reflection. This is the section's most important figure and it should be built, not merely shown.
• A redraw of Fig 5.10 (Part I p. 127) with the three bases labelled, for section 7. Optional if time is short; Fig 5.11 carries the argument.
• A two-box arrow diagram for section 7 with the domain and range labels exchanging places when the arrow reverses. Not in the book.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 5, §5.4 Exponential and Logarithmic Functions, the opening warning and the growth discussion with Fig 5.9, Part I pp. 125–126
• Definition 3 and the five features, Part I pp. 126–127; the natural base and Definition 4, Part I p. 127; the logarithm as a function with Fig 5.10, Part I p. 127
• The six features with Fig 5.11 and the two proved properties, Part I pp. 128–129
• Example 25 and Theorem 5 with its footnote, Part I p. 129; Example 26, Part I pp. 129–130
• Exercise 5.4, questions 1 to 10, Part I p. 130; Miscellaneous Example 39 part two, Part I p. 141; Summary, the two derivatives, Part I p. 146