S4 Mathematics: Indices and Logarithms prep guide | Index laws, log definition and exponential equations

2026-11-01

Tutor lesson-prep guides · Secondary Mathematics

Students know that a to the m times a to the n adds the indices, then start moving terms around at random when asked to solve three to the power 2x minus 1 equals 81. This lesson locks indices and logarithms together as one idea read in two directions.

Objectives — Students can: (1) use the three index laws confidently, including negative and fractional indices; (2) convert between a to the x equals b and x equals log base a of b; (3) solve equations of the form a to the x equals b by taking logarithms, and find growth or decay times.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Index notation	What does 2 cubed mean	Three twos multiplied
Linear equations	Solve 2x – 1 = 4	2.5
Calculator	Where are log and ln	log is base 10, ln is base e

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-12 min	Index laws revisited	2 cubed times 2 to the fourth is 2 to the seventh; 2 to the tenth is 1024	1024
12-26 min	Negative and fractional indices	a to minus n is 1 over a to the n	8 to the two thirds is 4
26-40 min	Definition of a logarithm	a to the x equals b means x is log base a of b	log base 2 of 7 is 2.807
40-50 min	Solving exponential equations	2 to the x equals 50 gives x as log 50 over log 2	5.64
50-56 min	Growth and decay	8 percent growth doubles, 5 percent decay halves	9.0 and 13.5 years
56-60 min	Exit check	Three questions collected	Done independently

03 Teaching script (read it out)

Scene 1: The index laws come from counting factors

2 cubed times 2 to the fourth is three twos then four twos, so seven twos in total and the indices add. Dividing cancels factors, so the indices subtract. A power of a power multiplies out, so the indices multiply.
A negative index means a reciprocal: 2 to the minus 3 is one over eight. A fractional index means a root: 8 to the one third is 2, so 8 to the two thirds is the cube root squared, which is 4. Denominator takes the root, numerator takes the power.
2 to the tenth equals 1024 is worth memorising; it comes back in logarithms and in binary.

Task: simplify 2 to the fifth times 2 to the minus second, all over 2 to the fourth, giving 2 to the minus first, which is 0.5, naming the law used at each step.

Scene 2: A logarithm is the question “what index?”

log base a of b asks how many times a must be multiplied by itself to reach b. So a to the x equals b and x equals log base a of b are the same sentence written two ways.
log base 2 of 8 is 3 because 2 cubed is 8; log base 10 of 1000 is 3; log base 5 of 1 is 0; log base 3 of one ninth is minus 2.
The laws translate straight across: the log of a product is the sum, the log of a quotient is the difference, and the log of a power brings the index down. Check: log 8 plus log 125 is log 1000, which is 3.
Change of base: log base 2 of 7 is ln 7 over ln 2, which is 2.807. Calculators only offer log and ln, so change of base has to be automatic.

Warning: the log of a sum is not the sum of the logs. Substitute 1 and 9 to see it: log 10 is 1, while log 1 plus log 9 is 0.954.

Scene 3: Solving exponential equations and word problems

For 2 to the x equals 50, take logs of both sides: x log 2 equals log 50, so x is log 50 over log 2, which is 5.64. For 3 to the power 2x minus 1 equals 81, first write 81 as 3 to the fourth, then compare indices: 2x minus 1 equals 4, so x is 2.5.
Growth problem: 5,000 at 6 percent compound interest reaching 8,000 means solving 1.06 to the n equals 1.6, so n is ln 1.6 over ln 1.06, which is 8.07, that is during the ninth year.
Doubling time: 1.08 to the n equals 2 gives 9.0 years. Half-life: 0.95 to the n equals 0.5 gives 13.5 years.
Three fixed steps: form the equation, take logs of both sides, divide and attach the unit.

Task: solve 5 to the x equals 12, giving 1.544, and explain why taking a square root of both sides does not help.

04 Question types and worked steps

Type	Example	Steps	Answer
Simplification	Index expressions	Add, subtract or multiply indices on the same base	2 to the minus 1 is 0.5
Definition	Evaluating logs	Rewrite as a to the x equals b	log base 3 of one ninth is minus 2
Equation	Solving a to the x equals b	Take logs, then divide	x is 5.64
Application	Growth and decay	Form, take logs, attach units	9.0 years to double

Sanity check (teach this): Always substitute back: 2 to the 5.64 is about 50 and 1.06 to the 8.07 is about 1.6. Check the domain: a logarithm only accepts positive arguments, so log of x minus 3 needs x above 3, and the base must be positive and not equal to one.
Check the context: a doubling time of 9.0 years may be reported as 9 years, while 0.9 years signals a decimal slip. Compounding periods round up, so 8.07 years means during the ninth year, never the eighth.

05 Independent practice (3 questions)

Question	Answer	What to watch
Simplify 2 to the fifth times 2 to the minus third, over 2 to the fourth	2 to the minus 2, that is 0.25	Add and subtract indices
log base 2 of 32	5	2 to the fifth is 32
Solve 3 to the power 2x minus 1 equals 81	x = 2.5	81 is 3 to the fourth
Years for 5,000 at 6 percent to reach 8,000	8.07, during year 9	ln 1.6 over ln 1.06

06 Common mistakes and what to say

Mistake	Student writes	Say this
Splitting the log of a sum	Writing log of x plus 2 as log x plus log 2	“Logs only split products and quotients. A plus inside cannot come out, and one and nine show it.”
Reading a negative index as a negative number	Writing 2 to the minus 3 as minus 8	“A negative index means reciprocal, not a minus sign: 2 to the minus 3 is one eighth.”
Taking logs when the bases already match	Taking logs of 3 to the 2x minus 1 equals 81 straight away	“If 81 is a power of 3, match the bases first. It is faster and far harder to get wrong.”
Rounding the number of periods down	Answering 8 years	“Short of the period means short of the money. Periods always round up.”

07 Homework (paper, section and question numbers)

Secondary mathematics chapter index — All chapters
Mathematics practice paper — Indices and logarithms, questions 8 to 14

08 Exit check (last 5 minutes)

Question	Answer	Decision
Value of log base 2 of 32	5	All correct: move to sequences
Value of 2 to the minus 3	0.125	Wrong: reteach negative indices
Solve 2 to the x equals 50	5.64	Two wrong: back to the definition

09 Tutor FAQ

Why do logarithms solve exponential equations?

Taking a logarithm brings the unknown index down into a factor, so the equation becomes linear in the unknown.

What is the difference between log and ln?

They differ only in base: log is base 10 and ln is base e, about 2.718. Change of base converts between any bases.

How does the growth rate set the doubling time?

Doubling time is ln 2 over ln of one plus the rate, so 8 percent gives about 9 years and 1 percent about 70 years, which is the rule of 72.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.