Further Logarithms
Numbers · Further Logarithms
Syllabus tag: KCSE | Mathematics | Form 3 | Topic 5 Further Logarithms
Lesson objectives
By the end of this topic, you should be able to:
- Derive the logarithmic relation from index form and vice versa.
- State and use the laws of logarithms to simplify expressions.
- Solve logarithmic equations.
- Apply logarithms to further computations.
Further Logarithms
A logarithm is an index. Knowing that makes every log law a restatement of an index law.
a) The relationship
aˣ = b and log_a b = x say exactly the same thing.
The logarithm is the index you raise the base to.
Reading it aloud helps: log₂ 8 = 3 says "the power of 2 that gives 8 is 3".
When no base is written, base 10 is understood. These are common logarithms.
b) The laws
Multiplying inside becomes adding outside, because indices add when powers multiply.
Dividing becomes subtracting.
A power inside becomes a multiplier outside.
Every one of these mirrors an index law exactly. They are not new rules.
c) Two useful special cases
log_a a = 1, since a¹ = a.
log_a 1 = 0, since a⁰ = 1.
So log 10 = 1 and log 1 = 0 for common logs.
d) What you cannot do
log(A + B) is not log A + log B.
The laws apply to products and quotients inside the logarithm, never to sums.
This mirrors the index laws, where aᵐ × aⁿ adds indices but aᵐ + aⁿ does not simplify.
e) Solving logarithmic equations
Combine the logs into a single logarithm using the laws.
If log A = log B with the same base, then A = B.
Solve the resulting equation.
Always check your answer. The logarithm of a negative number or of zero is undefined. Reject any root that makes an argument non-positive, even if the algebra allows it.
f) Change of base
Sometimes a different base is needed.
log_a b = log b ÷ log a, using common logs on the right.
This lets any logarithm be evaluated from base-10 tables or a calculator.
g) Solving index equations with logs
If the unknown is in the index, take logs of both sides.
Take 2ˣ = 50. Taking logs gives x log 2 = log 50. So x = log 50 ÷ log 2, about 5.64.
This is the main practical reason logarithms exist.
h) Where this is used
Compound interest, where time is the unknown. Population growth. pH and the Richter scale, both logarithmic. Any equation with the unknown as an exponent.
Words to know
- Logarithm -- the power to which a base must be raised to give a number.
- Base -- the number being raised to a power in the index form.
- Common logarithm -- a logarithm to base 10, written without a base.
- Change of base -- converting logₐ b into a quotient of base-10 logarithms.
- Extraneous root -- a solution produced by the algebra that the original equation rejects.
:::checkpoint Check yourself
- Write 3⁴ = 81 in logarithm form.
- Simplify log 8 + log 5 − log 4.
- Solve log x + log 3 = log 12.
- Why is log(A + B) not equal to log A + log B? :::
Bridge to practice
The exercises begin with converting between forms and the special values, move through the laws and simplification, and finish with equations, change of base and growth problems. For every logarithmic equation you solve, substitute each root back before writing your final answer.