Logarithms UPCAT Reviewer: Laws, Equations, and Applications

TEACHER ABI UPCAT MATHEMATICS

Logarithms: Foundations and Applications

Read logarithms as exponents, use the laws with meaning, and recognize logarithmic scales without relying on memorized symbol manipulation.

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Logarithms

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A logarithm answers an exponent question

The statement log_b a = c means: “To what exponent must b be raised to produce a?”

log_b a = c ⇔ b^c = a

For real logarithms, the base must satisfy b > 0 and b≠1, while the argument must be positive.

Product law

log_b(MN) = log_bM + log_bN

Quotient law

log_b(MN) = log_bM−log_bN

Power law

log_b(M^k) = klog_bM

Inverse relationship

log_b(b^x) = x and b^(log_bx) = x for valid values.

DO IT FAST

Ask: “What power makes this?”

1. Rewrite familiar numbers as powers of the base.

2. Convert between logarithmic and exponential form.

3. Combine logs only through products, quotients, and powers.

4. Check the domain after solving. Every logarithm argument must remain positive.

Why it works

This keeps logarithms connected to exponent meaning and prevents false rules such as splitting log(x + y).

WORKED EXAMPLES

Five forms you should recognize

1. Exact value

log₂32 = 5 because 2⁵ = 32

2. Negative value

log₁₀0.001 = −3 because 10⁻³ = 0.001

3. Combine logs

log₂x + log₂y−log₂4 = log₂(xy4)

4. Equation and domain

log₂x + log₂(x−2) = 3x(x−2) = 8 → x = 4 or −2

The domain x > 2 rejects −2.

5. Logarithmic scale

A difference of 2 on a base-10 logarithmic amplitude scale represents a factor of 10² = 100.

COMMON TRAPS

Check before you commit

• Treating log(x + y) as logx + logy
• Forgetting that logarithm arguments must be positive
• Confusing log_ba with a^b
• Dropping a coefficient when using the power law
• Accepting algebraic roots that violate the domain
• Treating a logarithmic scale as an ordinary linear scale

FIVE-FORM SKILL CHECK

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Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

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QUICK ANSWERS

Logarithms: Foundations and Applications FAQ

Does log without a written base mean base 10?

In most school algebra contexts, yes; ln specifically means base e.

Can a logarithm be negative?

Yes. For a base greater than 1, inputs between 0 and 1 have negative logarithms.

Why must the argument be positive?

No real exponent on a positive base produces zero or a negative number.

RELATED COMPETENCIES

Continue your mathematics review.

Exponent RulesScientific NotationFunctionsAcids and Bases

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