Logarithms Explained: Exponents, Rules & Worked Examples

Read the base, argument and result

In log₃81=4, 3 is the base, 81 is the argument and 4 is the exponent needed to reach it. Read the equation back as 3⁴=81. The logarithm is not “81 divided by 3”.

The base must be positive and different from 1. Real logarithms accept positive arguments. Zero and negative arguments are not allowed in this real-number lesson.

Connect zero and negative exponents

Since b⁰=1, log_b(1)=0. Since 2⁻³=1/8, log₂(1/8)=−3. A negative logarithm value is possible even though its argument must be positive.

For bases greater than 1, arguments between 0 and 1 produce negative logarithms. For bases between 0 and 1, the graph decreases; do not apply every increasing-base intuition to that case.

Derive the product rule

If A=bᵐ and B=bⁿ with positive A and B, then AB=bᵐ⁺ⁿ. Thus log_b(AB)=log_b(A)+log_b(B). The product inside becomes addition of exponents outside.

For base 2, 8×4=32, and log₂32=5=3+2. Both logarithms must use the same base. This rule does not say log_b(A+B)=log_b(A)+log_b(B).

Quotient and power rules

Division subtracts exponents, so log_b(A/B)=log_b(A)−log_b(B), for positive A and B. Also log_b(Aᵏ)=k log_b(A) for positive A and real k. These conditions protect the real logarithms used in the equality.

For example log₃(81/3)=4−1=3. With log₂(8²), the answer is 2×3=6. Do not drop brackets when only part of an expression is raised to a power.

Common logarithms, natural logarithms and calculators

A common logarithm uses base 10 and is often written log. A natural logarithm uses base e and is written ln. Check the notation in your course or calculator rather than assuming an omitted base in every context.

Change of base gives log_b(a)=ln(a)/ln(b). This is a ratio of two logs, not ln(a/b). When an exact power is visible, exact evaluation is clearer than a rounded calculator answer.

Solve an equation and keep the restrictions

For log₂(x−1)=3, convert to x−1=2³=8, giving x=9. Check x−1>0: it is 8. For an equation with several logarithms, write the domain conditions before combining them.

Algebra may generate values that violate those conditions. A candidate root is not an answer until every original logarithm is defined. A numerical check should use the original equation.

Read the graph and its inverse relationship

For b>1, y=log_b(x) increases, passes through (1,0) and (b,1), and approaches the vertical line x=0 without reaching it. Its real domain is x>0 and range is all real numbers.

The exponential y=bˣ has the coordinates reversed. Reflecting across y=x connects the two graphs. Different scales can distort the picture, so label axes and use the same unit scale when illustrating reflection.

Use logarithms as a reasoning tool

A logarithm helps solve for an exponent, compare multiplicative scales and represent wide ranges. It is not automatically appropriate for an additive pattern. For this introductory guide, focus on interpreting the exponent before memorising laws.

Keep a record with four entries: base and domain; equivalent exponential equation; calculated value; substitution check. If a rule seems uncertain, test it with simple powers that you can verify independently.

See the reasoning

Worked examples and checks

Try, explain, check

12 questions with explained answers

Try each task before opening its explanation. These are original BananaLearn questions, not official exam or past-paper questions. Where several answers can work, the explanation gives the criteria.

1. Evaluate log₂32.

   Check answer and reasoning

   5
   2⁵=32.

2. Evaluate log₄1.

   Check answer and reasoning

   0
   4⁰=1.

3. Evaluate log₃(1/9).

   Check answer and reasoning

   −2
   3⁻²=1/9.

4. Is log₂0 defined over the real numbers?

   Check answer and reasoning

   No
   A positive base raised to a real power is never zero.

5. Is log₁7 valid?

   Check answer and reasoning

   No
   Base 1 is excluded.

6. Expand log₅(AB) for A,B>0.

   Check answer and reasoning

   log₅A+log₅B
   Multiplication adds the exponents.

7. Simplify log₂16−log₂4.

   Check answer and reasoning

   2
   4−2=2, also log₂(16/4)=log₂4.

8. Evaluate log₁₀(1000²).

   Check answer and reasoning

   6
   1000²=10⁶.

9. Solve log₅(x+1)=2.

   Check answer and reasoning

   24
   x+1=25 and the argument is positive.

10. Give the domain of log₃(2x−6).

    Check answer and reasoning

    x>3
    Its argument must be strictly positive.

11. Does a negative log answer require a negative argument?

    Check answer and reasoning

    No
    For example log₂(1/2)=−1.

12. Write log₇11 using ln.

    Check answer and reasoning

    ln(11)/ln(7)
    The denominator records the base; it is not ln(11/7).

A checking record to use in your notebook

Copy these headings and fill them for one worked task, then for a fresh attempt. Use your own evidence rather than copying the model answer.

Question or sentence

What am I being asked to explain?

Model or evidence

Which diagram, rule or passage detail supports my method?

Work

What steps connect the question to the answer?

Independent check

Can I substitute, recount, trace a path or check the text?

Next attempt

What changes in a fresh example?

Choose the next useful step

• Revisit the exponent laws
• Understand inverse rules and domains
• Algebra 2 support for exponential and logarithmic ideas

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Sources, level and update

Original BananaLearn explanations, examples, illustrations and practice. Source facts checked 7 November 2026. Suggested levels are learning guides, not a claim that every board or textbook uses the same sequence.

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