Properties of Logarithms

Introduction to the Properties of Logarithms

Logarithms provide a powerful way to work with exponential expressions. While exponents describe repeated multiplication, logarithms determine the exponent required to produce a particular number.

The properties of logarithms allow complicated logarithmic expressions to be rewritten in simpler forms.

These properties are especially useful when:

• Simplifying mathematical expressions.

• Solving exponential and logarithmic equations.

• Combining several logarithms into a single expression.

• Expanding logarithmic expressions into separate terms.

• Working with scientific and engineering formulas.

• Analysing exponential growth and decay.

Three fundamental properties form the foundation of logarithmic algebra:

• Product Rule: Multiplication becomes addition.

• Quotient Rule: Division becomes subtraction.

• Power Rule: Exponents become multipliers.

These rules follow directly from the laws of exponents.

Reviewing the Meaning of a Logarithm

Recall the relationship:

aᵇ = c ⇔ logₐ(c) = b

where:

• a is the base.

• b is the exponent.

• c is the result.

For real logarithms:

• a > 0.

• a ≠ 1.

• c > 0.

Example

Consider:

2⁵ = 32

The equivalent logarithmic statement is:

log₂(32) = 5

The logarithm tells us which exponent produces the argument.

This connection to exponents explains why logarithmic rules work.

The Product Rule

The product rule states that the logarithm of a product equals the sum of the logarithms of its factors.

logₐ(MN) = logₐ(M) + logₐ(N)

where:

• a > 0 and a ≠ 1.

• M > 0.

• N > 0.

Understanding the Rule

Suppose:

2³ = 8

and:

2⁴ = 16

Multiplying:

8 × 16 = 128

Using exponent laws:

2³ × 2⁴ = 2⁷

Therefore:

log₂(128) = 7

But:

log₂(8) = 3

and:

log₂(16) = 4

So:

log₂(8 × 16) = log₂(8) + log₂(16)

7 = 3 + 4

The product rule works because multiplying powers with the same base adds their exponents.

Deriving the Product Rule

Let:

logₐ(M) = p

and:

logₐ(N) = q

Convert to exponential form:

M = aᵖ

N = aᑫ

Multiply:

MN = aᵖ × aᑫ

Using the exponent law:

aᵖ × aᑫ = a^(p+q)

Therefore:

MN = a^(p+q)

Convert back to logarithmic form:

logₐ(MN) = p + q

Substitute the original logarithms:

logₐ(MN) = logₐ(M) + logₐ(N)

This derivation shows that the product rule is a direct consequence of the laws of exponents.

Applying the Product Rule

Worked Example: Evaluating a Product

Evaluate:

log₂(4) + log₂(8)

Apply the product rule:

= log₂(4 × 8)

= log₂(32)

Since:

2⁵ = 32

Answer = 5

Worked Example: Combining Logarithms

Simplify:

log₃(5) + log₃(9)

Apply the product rule:

= log₃(5 × 9)

= log₃(45)

The result is a single logarithmic expression.

Worked Example: Three Logarithms

Condense:

log₂(3) + log₂(5) + log₂(7)

Combine the arguments:

= log₂(3 × 5 × 7)

= log₂(105)

The product rule can be extended to any finite number of positive factors.

Expanding Products

The product rule can also be used in reverse.

Consider:

log₅(6x)

Assuming x > 0:

log₅(6x) = log₅(6) + log₅(x)

This process is called expanding a logarithmic expression.

Worked Example

Expand:

log₂(12x)

Factorise:

12x = 3 × 4 × x

Apply the product rule:

log₂(12x) = log₂(3) + log₂(4) + log₂(x)

Since:

log₂(4) = 2

Answer = log₂(3) + 2 + log₂(x)

This expansion is valid for x > 0.

The Quotient Rule

The quotient rule states that the logarithm of a quotient equals the difference between the logarithms of the numerator and denominator.

logₐ(M/N) = logₐ(M) − logₐ(N)

where:

• a > 0 and a ≠ 1.

• M > 0.

• N > 0.

Understanding the Rule

Consider:

2⁶ = 64

and:

2² = 4

Dividing:

64/4 = 16

Using exponent laws:

2⁶/2² = 2⁴

Therefore:

log₂(16) = 4

But:

log₂(64) = 6

and:

log₂(4) = 2

Thus:

log₂(64/4) = log₂(64) − log₂(4)

4 = 6 − 2

Division of powers with the same base corresponds to subtraction of exponents.

Deriving the Quotient Rule

Let:

logₐ(M) = p

logₐ(N) = q

Then:

M = aᵖ

N = aᑫ

Divide:

M/N = aᵖ/aᑫ

Using exponent laws:

M/N = a^(p−q)

Convert to logarithmic form:

logₐ(M/N) = p − q

Therefore:

logₐ(M/N) = logₐ(M) − logₐ(N)

Applying the Quotient Rule

Worked Example: Evaluating a Difference

Evaluate:

log₃(81) − log₃(9)

Apply the quotient rule:

= log₃(81/9)

= log₃(9)

Since:

3² = 9

Answer = 2

Worked Example: Combining Expressions

Condense:

log₅(20) − log₅(4)

= log₅(20/4)

= log₅(5)

Answer = 1

Worked Example: Expanding a Quotient

Expand:

log₂(x/8)

Assuming x > 0:

= log₂(x) − log₂(8)

Since:

log₂(8) = 3

Answer = log₂(x) − 3

The Power Rule

The power rule states that an exponent inside a logarithm can be moved in front as a multiplier.

logₐ(Mᵏ) = k logₐ(M)

where a > 0, a ≠ 1, M > 0, and k is any real number.

Understanding the Rule

Consider:

log₂(8³)

Since:

8 = 2³

then:

8³ = (2³)³

= 2⁹

Therefore:

log₂(8³) = 9

Using the power rule:

3log₂(8)

= 3 × 3

= 9

Both methods produce the same answer.

Deriving the Power Rule

Let:

logₐ(M) = p

Then:

M = aᵖ

Raise both sides to the power k:

Mᵏ = (aᵖ)ᵏ

Using exponent laws:

Mᵏ = a^(pk)

Convert to logarithmic form:

logₐ(Mᵏ) = pk

Since:

p = logₐ(M)

we obtain:

logₐ(Mᵏ) = k logₐ(M)

Applying the Power Rule

Worked Example: Expanding a Power

Expand:

log₃(x⁴)

For x > 0:

= 4log₃(x)

Worked Example: Evaluating an Expression

Evaluate:

3log₂(4)

Since:

log₂(4) = 2

3log₂(4) = 3 × 2

Answer = 6

Alternatively:

3log₂(4) = log₂(4³)

= log₂(64)

= 6

Worked Example: Fractional Exponents

Expand:

log₅(√x)

Assuming x > 0:

√x = x^(1/2)

Therefore:

log₅(√x) = log₅(x^(1/2))

= (1/2)log₅(x)

Worked Example: Negative Exponents

Simplify:

log₂(1/x³)

Assuming x > 0:

1/x³ = x⁻³

Therefore:

log₂(x⁻³) = −3log₂(x)

Answer = −3log₂(x)

Important Special Cases of the Power Rule

The power rule leads to several useful identities.

Logarithm of 1

Since:

a⁰ = 1

logₐ(1) = 0

Logarithm of the Base

Since:

a¹ = a

logₐ(a) = 1

Logarithm of a Power of the Base

logₐ(aᵏ) = k

For example:

log₃(3⁷) = 7

Exponential and Logarithmic Inverses

a^(logₐ(x)) = x

for x > 0.

These relationships make many logarithmic calculations easier.

Summary of the Three Main Rules

For the product and quotient rules, M and N must be positive. For the power rule as written, M must be positive.

The logarithms must also use the same valid base.

Expanding Logarithmic Expressions

Expanding means rewriting a logarithm of a product, quotient, or power as separate logarithmic terms.

The usual approach is:

1. Identify products and quotients.

2. Apply the product and quotient rules.

3. Apply the power rule.

4. Simplify any logarithms that can be evaluated.

Worked Example: Product and Power

Expand completely:

log₂(8x³)

Assume x > 0.

Apply the product rule:

= log₂(8) + log₂(x³)

Apply the power rule:

= log₂(8) + 3log₂(x)

Since:

log₂(8) = 3

Answer = 3 + 3log₂(x)

Worked Example: Product, Quotient, and Power

Expand:

log₃(9x²/y⁴)

Assume x > 0 and y > 0.

Apply the quotient rule:

= log₃(9x²) − log₃(y⁴)

Apply the product rule:

= log₃(9) + log₃(x²) − log₃(y⁴)

Apply the power rule:

= 2 + 2log₃(x) − 4log₃

Final Answer:

2 + 2log₃(x) − 4log₃

Worked Example: Square Roots and Quotients

Expand:

ln(√x/y³)

Assume x > 0 and y > 0.

Rewrite the square root:

√x = x^(1/2)

Apply the quotient rule:

= ln(x^(1/2)) − ln(y³)

Apply the power rule:

= (1/2)ln(x) − 3ln

Handling Variables That May Be Negative

When expanding logarithms involving even powers, domain restrictions require special care.

For example:

log₂(x²)

is defined whenever x ≠ 0, including negative x.

Therefore, the expansion valid over the full real domain is:

log₂(x²) = 2log₂|x|

It is not generally correct to write 2log₂(x), because log₂(x) is undefined when x is negative.

Similarly:

ln(x²y⁴) = 2ln|x| + 4ln|y|

for x ≠ 0 and y ≠ 0.

Absolute values are unnecessary when the problem explicitly states that all variables are positive.

This distinction becomes especially important in advanced algebra and calculus.

Condensing Logarithmic Expressions

Condensing is the reverse of expanding.

It involves combining several logarithmic terms into a single logarithm.

The general procedure is:

1. Use the power rule to move coefficients into exponents.

2. Combine addition using the product rule.

3. Combine subtraction using the quotient rule.

4. Simplify the argument where possible.

Worked Example: Condensing Two Terms

Condense:

log₂(x) + log₂(5)

Apply the product rule:

= log₂(5x)

Valid for x > 0.

Worked Example: Coefficients and Subtraction

Condense:

3log₄(x) − 2log₄

Assume x > 0 and y > 0.

Apply the power rule:

= log₄(x³) − log₄(y²)

Apply the quotient rule:

= log₄(x³/y²)

Worked Example: Combining Three Terms

Condense:

2ln(x) + ln − 3ln(z)

Assume x, y, and z are positive.

Apply the power rule:

= ln(x²) + ln − ln(z³)

Apply the product rule:

= ln(x²y) − ln(z³)

Apply the quotient rule:

= ln(x²y/z³)

Multi-Step Expansion and Condensation

Some problems require all three rules.

Worked Example: Complete Expansion

Expand:

log₅[25x³√y/z²]

Assume x > 0, y > 0, and z > 0.

Step 1: Apply the Quotient Rule

= log₅(25x³√y) − log₅(z²)

Step 2: Apply the Product Rule

= log₅(25) + log₅(x³) + log₅(√y) − log₅(z²)

Step 3: Apply the Power Rule

= log₅(25) + 3log₅(x) + (1/2)log₅ − 2log₅(z)

Step 4: Evaluate the Constant

Since:

5² = 25

log₅(25) = 2

Final Answer

2 + 3log₅(x) + (1/2)log₅ − 2log₅(z)

Reverse Process

Starting with the expanded expression, the same rules can be applied in reverse to recover the original single logarithm.

This demonstrates that expansion and condensation are inverse algebraic processes.

Simplifying Logarithmic Expressions

Logarithmic expressions can often be simplified without a calculator.

Worked Example

Simplify:

log₂(16) + log₂(8) − log₂(4)

Apply the product and quotient rules:

= log₂[(16 × 8)/4]

= log₂(32)

Answer = 5

Worked Example

Simplify:

2log₃(9) − log₃(27)

Since:

log₃(9) = 2

log₃(27) = 3

we obtain:

2(2) − 3

Answer = 1

Alternatively:

= log₃(9²/27)

= log₃(3)

= 1

The Change-of-Base Formula

The change-of-base formula allows logarithms to be rewritten using a different base.

logₐ(x) = log(x)/log(a)

or:

logₐ(x) = ln(x)/ln(a)

where a > 0, a ≠ 1, and x > 0.

Worked Example

Evaluate:

log₄(20)

Using common logarithms:

log₄(20) = log(20)/log(4)

≈ 1.30103/0.60206

≈ 2.161

Why Change of Base Works

Let:

y = logₐ(x)

Then:

aʸ = x

Take the natural logarithm of both sides:

ln(aʸ) = ln(x)

Apply the power rule:

y ln(a) = ln(x)

Rearrange:

y = ln(x)/ln(a)

Therefore:

logₐ(x) = ln(x)/ln(a)

The change-of-base formula is itself an application of the power rule.

Simplifying Expressions with Different Bases

The product and quotient rules require logarithms with the same base.

However, the change-of-base formula can help when different bases appear.

Worked Example

Simplify:

log₂(8) + log₄(16)

Evaluate each logarithm:

log₂(8) = 3

log₄(16) = 2

Therefore:

Answer = 5

We cannot directly combine these terms using the product rule because their bases are different.

Another Example

Evaluate:

log₂(3) × log₃(8)

Using change of base:

log₂(3) = ln(3)/ln(2)

log₃(8) = ln(8)/ln(3)

Multiply:

= [ln(3)/ln(2)] × [ln(8)/ln(3)]

Cancel ln(3):

= ln(8)/ln(2)

= log₂(8)

Answer = 3

This illustrates how logarithmic properties can reveal relationships that are not immediately obvious.

Applying Logarithm Rules to Exponential Equations

Logarithmic properties are essential for solving exponential equations.

Worked Example

Solve:

3^(2x) = 50

Step 1: Take Logarithms

ln(3^(2x)) = ln(50)

Step 2: Apply the Power Rule

2x ln(3) = ln(50)

Step 3: Isolate x

x = ln(50)/[2ln(3)]

Step 4: Calculate

x ≈ 1.780

Verification

3^(2 × 1.780) ≈ 50

The result is reasonable.

Applying Logarithm Rules to Growth Models

Logarithms appear in many scientific models.

Consider exponential population growth:

P(t) = P₀e^(kt)

where:

• P(t) is the population at time t.

• P₀ is the initial population.

• k is the continuous growth rate.

• t is time.

Worked Example

A population follows:

P(t) = 500e^(0.08t)

Find when the population reaches 2000.

Step 1: Substitute

2000 = 500e^(0.08t)

Step 2: Divide

4 = e^(0.08t)

Step 3: Take Natural Logarithms

ln(4) = ln(e^(0.08t))

Step 4: Apply the Power Rule

ln(4) = 0.08t

Step 5: Solve

t = ln(4)/0.08

t ≈ 17.33

The population reaches 2000 after approximately 17.33 time units.

Logarithmic Properties in Science

The properties of logarithms are useful because they convert multiplication and powers into simpler algebraic operations.

Sound Intensity

Sound intensity level is calculated using:

L = 10log₁₀(I/I₀)

Applying the quotient rule:

L = 10[log₁₀(I) − log₁₀(I₀)]

when I and I₀ are expressed in the same positive units.

This demonstrates how a logarithmic ratio can be expanded into a difference.

Radioactive Decay

A simplified radioactive decay model is:

N(t) = N₀e^(−λt)

Taking natural logarithms:

ln[N(t)/N₀] = −λt

This form can be used to calculate decay constants or elapsed time.

Chemistry

The pH scale uses a logarithmic relationship:

pH = −log₁₀(a_H⁺)

where a_H⁺ is the dimensionless hydrogen-ion activity.

The power and quotient rules help analyse concentration ratios and changes in acidity.

Information Theory

Logarithms are used to quantify information.

For an event with probability p:

I = −log₂(p)

where I is the information content in bits.

For independent events with probabilities p and q:

−log₂(pq) = −log₂(p) − log₂(q)

Thus, the information associated with independent events adds.

Recognising Invalid Logarithm Rules

Not every algebraic operation has a corresponding logarithmic property.

The following statements are generally incorrect.

Incorrect:

logₐ(M + N) = logₐ(M) + logₐ(N)

Incorrect:

logₐ(M − N) = logₐ(M) − logₐ(N)

Incorrect:

logₐ(MN) = logₐ(M) × logₐ(N)

Numerical Counterexample

Consider:

log₂(2 + 2)

= log₂(4)

= 2

But:

log₂(2) + log₂(2)

= 1 + 1

= 2

These happen to agree, but this does not prove a general rule.

Choose a different example:

log₂(2 + 4)

= log₂(6)

≈ 2.585

However:

log₂(2) + log₂(4)

= 1 + 2

= 3

The values are different.

Therefore:

logₐ(M + N) ≠ logₐ(M) + logₐ(N)

in general.

The product rule applies to multiplication, not addition.

Choosing Between Expansion and Condensation

Different problems benefit from different forms.

A useful strategy is to choose the form that makes the next algebraic step easiest.

Common Mistakes

Confusing multiplication with addition

The product rule applies to multiplication inside a logarithm, not addition.

Confusing division with subtraction

The quotient rule applies to a quotient inside a logarithm, not to subtraction inside the argument.

Applying the power rule incorrectly

The exponent becomes a multiplier outside the logarithm.

Using different bases in the product or quotient rule

The logarithms must have the same base before they can be combined directly.

Forgetting domain restrictions

Real logarithms require positive arguments.

Ignoring absolute values when expanding even powers

For example, logₐ(x²) = 2logₐ|x| for x ≠ 0.

Applying the product rule to negative factors

The real product rule requires both factors to be positive, even if their product is positive.

Forgetting to distribute coefficients

For example:

2[log(x) + log] = 2log(x) + 2log

Condensing before applying the power rule

It is often easier to move coefficients into exponents first.

Using logarithmic rules without checking equivalence

Algebraic transformations must preserve the original domain.

Rounding intermediate calculations too early

Retain sufficient precision until the final answer.

Key Terms

• Logarithm: The exponent required to raise a given base to a specified positive value.

• Base: The positive number, other than 1, used in a logarithmic expression.

• Argument: The positive expression inside a logarithm.

• Product Rule: The property that converts the logarithm of a product into a sum of logarithms.

• Quotient Rule: The property that converts the logarithm of a quotient into a difference of logarithms.

• Power Rule: The property that allows an exponent inside a logarithm to become a multiplier.

• Expansion: Rewriting one logarithm as a sum or difference of simpler logarithms.

• Condensation: Combining multiple logarithmic terms into one logarithm.

• Logarithmic Expression: An algebraic expression containing logarithms.

• Logarithmic Identity: A relationship involving logarithms that is valid throughout its stated domain.

• Exponential Form: A mathematical statement written using a base raised to an exponent.

• Logarithmic Form: An equivalent statement expressing the exponent as a logarithm.

• Inverse Operation: An operation that reverses another operation.

• Common Logarithm: A logarithm with base 10.

• Natural Logarithm: A logarithm with base e.

• Change-of-Base Formula: A formula that rewrites a logarithm using a different base.

• Exponent Law: A mathematical rule describing operations involving powers.

• Domain Restriction: A condition limiting allowable input values.

• Absolute Value: The nonnegative magnitude of a real number.

• Equivalent Expressions: Expressions that have equal values for every input in their common domain.

• Exponential Equation: An equation containing a variable in an exponent.

• Logarithmic Equation: An equation containing one or more logarithmic expressions.

• Exponential Growth: Growth by a constant multiplicative factor over equal time intervals.

• Exponential Decay: Decrease by a constant multiplicative factor over equal time intervals.

• Continuous Growth: Exponential growth modelled using the constant e.

• Mathematical Simplification: Rewriting an expression in a more useful equivalent form.

Key Takeaways

• The three fundamental logarithm properties are the product rule, quotient rule, and power rule.

• The product rule converts multiplication inside a logarithm into addition outside it.

• The quotient rule converts division inside a logarithm into subtraction outside it.

• The power rule converts an exponent inside a logarithm into a multiplier.

• These rules follow directly from the laws of exponents.

• Logarithms can be expanded into sums and differences of simpler logarithms.

• Logarithmic expressions can also be condensed into a single logarithm.

• Expansion and condensation are reverse processes.

• Applying the power rule before condensing often simplifies the calculation.

• Logarithmic properties can be used to evaluate expressions without a calculator.

• The product and quotient rules require logarithms with the same base.

• The arguments of real logarithms must be positive.

• Absolute values may be necessary when expanding logarithms involving even powers of variables.

• The change-of-base formula allows logarithms to be evaluated using common or natural logarithms.

• Logarithmic rules are essential for solving exponential equations.

• The properties of logarithms simplify mathematical models involving growth, decay, sound intensity, and scientific measurements.

• There is no general logarithm rule that separates addition or subtraction inside an argument.

• A valid logarithmic simplification must preserve mathematical equivalence and domain restrictions.

• Mastering logarithmic properties provides the algebraic foundation for logarithmic equations, exponential modelling, and advanced calculus.

Check Your Understanding

Recall and Understanding

1. State the product, quotient, and power rules of logarithms.

2. Explain why the product rule follows from the laws of exponents.

3. Describe the difference between expanding and condensing logarithmic expressions.

4. Explain why logarithms must have the same base to use the product or quotient rule directly.

5. Explain why log₂(x²) cannot always be expanded as 2log₂(x) for every nonzero real x.

Application

6. Expand log₃(27x²), assuming x > 0.

7. Expand ln(x³/y²), assuming x > 0 and y > 0.

8. Condense 2log₅(x) + 3log₅, assuming x > 0 and y > 0.

9. Simplify log₂(32) + log₂(4) − log₂(8).

10. Condense 3ln(a) − 2ln(b) + ln(c), assuming all variables are positive.

Analysis and Challenge

11. Expand log₂[8x⁴√y/z³] completely, assuming x, y, and z are positive.

12. Condense 2log₃(x) + (1/2)log₃ − 4log₃(z) into a single logarithm, assuming all variables are positive.

13. Prove the change-of-base formula using the power rule.

14. Solve 5^(3x−1) = 40 using logarithms, giving the answer to three decimal places.

15. A population follows P(t) = 1200e^(0.06t). Determine when it will reach 5000 and explain how the power rule is used in the solution.

Final Summary

The properties of logarithms provide a systematic way to simplify and manipulate expressions involving logarithmic functions.

The product rule transforms multiplication into addition.

The quotient rule transforms division into subtraction.

The power rule transforms exponents into multipliers.

These properties are derived from the laws of exponents and are fundamental to logarithmic algebra.

By applying the rules in reverse, several logarithmic terms can be combined into a single expression.

The same properties can be used to expand complicated logarithmic expressions into simpler components.

Logarithmic properties are also essential for solving exponential equations, working with growth and decay models, and interpreting scientific relationships.

Careful attention must be given to bases, positive arguments, domain restrictions, and valid algebraic transformations.

The central principle is that logarithms convert multiplicative and exponential relationships into additive and algebraic relationships, making complex expressions easier to simplify, analyse, and solve.