Additional Mathematics Synthesis Guide 33: Exponential and Logarithmic Functions — Laws, Change of Base, Inverse Graphs, Equations and Models

Power relationship → inverse logarithm → apply laws legally → solve exactly → check domain → interpret the model.

1. Exponential and logarithmic statements are equivalent

For a>0, a≠1,

aˣ=y ⇔ log_ay=x, with y>0.

For example, 2⁵=32 is equivalent to log₂32=5.

The logarithm is not an unrelated operation. It is the inverse question to exponentiation.

2. Why logarithms require positive arguments

For a positive base a≠1, aˣ is always positive for real x. Therefore its inverse logarithm receives only positive outputs:

log_ax is defined for x>0.

Any logarithmic equation must preserve this domain condition throughout the solution.

3. Product law from exponent addition

Because aᵐaⁿ=a^m+n, logarithms satisfy

log_a(MN)=log_aM+log_aN

for M>0,N>0.

The law comes from exponent structure; it is not an arbitrary symbolic rule.

4. Quotient and power laws

• log_a(M/N)=log_aM−log_aN
• log_a(Mᵏ)=k log_aM

Each requires the logarithmic arguments to remain legal. The laws transform a valid logarithmic expression; they do not make a non-positive argument acceptable.

5. A common false law

In general,

log(M+N) ≠ logM+logN.

For base 10, log(10+90)=2, while log10+log90≈2.954. Product structure, not addition structure, creates the sum of logarithms.

6. Change of base

For positive valid bases a,b≠1,

log_ax = log_bx / log_ba.

Using natural logarithms:

log_ax=lnx/lna.

This allows a calculator with ln or log keys to evaluate logarithms in any valid base.

7. Deriving change of base

Let y=log_ax. Then aʸ=x. Taking ln:

y lna=lnx.

Hence y=lnx/lna.

The formula is simply the exponential equation solved using a different logarithmic language.

8. Inverse graphs

The graphs of y=aˣ and y=log_ax are reflections of each other in y=x.

• Exponential domain: all real x; range: y>0.
• Logarithmic domain: x>0; range: all real y.
• Exponential horizontal asymptote y=0 becomes logarithmic vertical asymptote x=0.
• The point (0,1) becomes (1,0).

Inverse functions exchange coordinates, domains and ranges.

9. Base greater than one

If a>1, y=aˣ is increasing and y=log_ax is also increasing.

For example, 2ˣ increases as x grows. Its inverse log₂x also increases: larger positive inputs require larger exponents.

10. Base between zero and one

If 0<a<1, both y=aˣ and y=log_ax are decreasing.

For example, (1/2)ˣ falls as x increases. The inverse logarithm to base 1/2 is correspondingly decreasing.

11. Solving exponential equations by matching bases

Solve

9ˣ=27.

Write both as powers of 3:

3^2x=3³.

Since the exponential function with base 3 is one-to-one, 2x=3, so x=3/2.

12. Solving when bases do not match

Solve 5ˣ=12.

Take ln:

x ln5=ln12

so

x=ln12/ln5.

This exact logarithmic form should be preserved until a decimal is requested.

13. Exponential equations can hide a quadratic

Solve

4ˣ−5·2ˣ+4=0.

Let u=2ˣ>0. Then 4ˣ=u²:

u²−5u+4=0=(u−1)(u−4).

So u=1 or4, giving x=0 or2.

The positivity condition u>0 belongs to the substitution.

14. Solving logarithmic equations by combining logs

Solve

ln(x−1)+ln(x+1)=ln8.

Domain: x−1>0 and x+1>0, so x>1.

Combine:

ln[(x−1)(x+1)]=ln8.

One-to-one logarithms give x²−1=8, so x=±3. Domain leaves x=3 only.

15. Domain rejection is not optional

An algebraic solution to a transformed logarithmic equation is only a candidate. Every logarithm argument in the original equation must remain positive.

Domain restrictions are part of the problem, not a final cosmetic check.

16. Exponential and logarithmic intersections

An equation such as aˣ=mx+c asks where an exponential graph meets a line.

Depending on parameters, there may be zero, one or more intersections. Algebra, graph behaviour and sometimes calculus can work together to justify the number of solutions.

Do not assume every exponential equation has a closed-form school-level solution.

17. Growth and decay models

A basic exponential model can be written

Q(t)=Q₀e^kt.

• k>0: growth.
• k<0: decay.
• Q₀=Q(0): initial value.

The model assumes proportional-rate behaviour. It should not be used beyond contexts where that assumption is reasonable.

18. Recovering a growth constant

Suppose Q(0)=100 and Q(3)=160 in the model Q=100e^kt.

Then

160=100e^3k

so e^3k=1.6 and

k=(ln1.6)/3.

The exact log form can be used in later predictions without premature rounding.

19. Doubling and halving times

For Q=Q₀e^kt, doubling time T satisfies e^kT=2, so

T=ln2/k

when k>0.

For decay k<0, a half-life H satisfies e^kH=1/2, giving

H=ln(1/2)/k=−ln2/k.

20. Linearising an exponential relationship

For y=Abˣ with A>0,b>0, take logs:

lny=lnA+x lnb.

A plot of lny against x is linear with gradient lnb and intercept lnA.

Guide 18 develops linear-law parameter recovery in greater depth.

21. Common failure patterns

• Using log(M+N)=logM+logN.
• Forgetting that logarithm arguments must be positive.
• Using base 1 or a non-positive logarithm base.
• Rounding logarithmic values before the final stage.
• Forgetting positivity after substituting u=aˣ.
• Solving a transformed polynomial but not checking original log domains.
• Confusing the exponential asymptote y=0 with the logarithmic asymptote x=0.
• Treating an exponential model as automatically valid outside its modelling assumptions.

22. A reliable exponential-logarithmic routine

1. Identify the base, domain and whether the structure is exponential or logarithmic.
2. Rewrite using common bases when that creates exact exponent comparison.
3. Otherwise take a convenient logarithm and solve symbolically.
4. Use log laws only on legal positive arguments.
5. For repeated exponential powers, consider a substitution such as u=aˣ with u>0.
6. For logarithmic equations, record domain restrictions before combining logs.
7. Verify solutions in the original equation.
8. For models, interpret constants and check whether the context supports exponential behaviour.

23. Practice set

1. Rewrite 3⁴=81 as a logarithmic statement.
2. Evaluate log₂32.
3. Simplify ln(ab)−lna.
4. Express log₅7 using natural logarithms.
5. Solve 8ˣ=4.
6. Solve 3ˣ=10 exactly.
7. Solve 9ˣ=27.
8. Solve 4ˣ−5·2ˣ+4=0.
9. Solve ln(x−2)=ln5.
10. Solve ln(x−1)+ln(x+1)=ln8.
11. State the domain of ln(3−x).
12. State the range of y=eˣ.
13. State the domain of y=lnx.
14. Describe the relationship between y=eˣ and y=lnx.
15. For Q=200e^0.1t, state Q(0).
16. For Q=Q₀e^kt, derive doubling time.
17. A quantity falls from 80 to 40 under Q=80e^kt. Express the half-life relation.
18. Linearise y=Abˣ.
19. Why can an algebraic root from a log equation be rejected?
20. Why is change of base useful?

Answers

1. log₃81=4.
2. 5.
3. lnb, for a,b>0.
4. ln7/ln5.
5. 2^3x=2², so x=2/3.
6. x=ln10/ln3.
7. x=3/2.
8. Let u=2ˣ: u²−5u+4=0, so u=1 or4; x=0 or2.
9. x−2=5, so x=7.
10. x=3 after domain x>1.
11. x<3.
12. y>0.
13. x>0.
14. They are inverse functions and reflections in y=x.
15. 200.
16. 2=e^kT, so T=ln2/k.
17. 1/2=e^kH, so H=ln(1/2)/k.
18. lny=lnA+xlnb.
19. It may make an original logarithm argument non-positive.
20. It converts any valid logarithm base into a calculator-supported or analytically convenient base.

24. What mastery looks like

Mastery means the learner sees logarithms as inverse exponent questions, derives or applies laws with domain control, chooses between matching bases and taking logarithms, recognises hidden quadratics in exponential equations, and interprets graph or model parameters instead of treating them as disconnected constants.

The transfer test is to mix exact equations, graph questions and modelling data without naming the method. If the learner can select the representation that exposes the inverse relationship and verify the result in the original domain, exponential-logarithmic reasoning is under control.

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Continue through Batch 09

• Guide 34: Trigonometric Identity Engineering — Addition Formulae, Double Angles, Simplification, Proof and Equation Reduction
• Guide 35: Kinematics from Calculus — Displacement, Velocity, Acceleration, Direction Changes and Total Distance
• Guide 36: Graph-to-Equation Reconstruction — Quadratics, Exponentials, Trigonometric Functions and Parameter Recovery
• Additional Mathematics Directory
• BTT Mathematics Hub