Laws of Logarithms

Master the multiplication, division, and power laws to simplify and manipulate logarithmic expressions

SPEC 6.4 • EXPONENTIALS AND LOGARITHMS

The Three Laws of Logarithms

These three laws allow us to manipulate and simplify logarithmic expressions:

Law 1: Multiplication

lo g_{a} (x y) = lo g_{a} x + lo g_{a} y

Log of product = sum of logs

Law 2: Division

lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y

Log of quotient = difference of logs

Law 3: Power

lo g_{a} (x^{n}) = n lo g_{a} x

Log of power = power × log

Critical Rule

All logarithms must have the SAME BASE to use these laws!

You cannot combine

lo g_{2} x + lo g_{3} y

- the bases are different.

Law 1: Multiplication Law

Law 1

lo g_{a} (x y) = lo g_{a} x + lo g_{a} y

In words: The log of a product equals the sum of the logs.

Example

lo g_{2} (8 \times 4) = lo g_{2} 8 + lo g_{2} 4 = 3 + 2 = 5

Check:

8 \times 4 = 32 = 2^{5}

✓

Remember

Multiplication INSIDE the log becomes addition OUTSIDE.

Both logs must have the same base!

Worked Example01

Expand $lo g_{2} (8 x^{3})$ using the laws of logarithms.

•

1. Apply multiplication law:

lo g_{2} (8 x^{3}) = lo g_{2} 8 + lo g_{2} (x^{3})

•

2. Apply power law to the second term:

= lo g_{2} 8 + 3 lo g_{2} x

•

3. Simplify

lo g_{2} 8 = 3

since

2^{3} = 8

Answer: $3 + 3 lo g_{2} x$

Law 2: Division Law

Law 2

lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y

In words: The log of a quotient equals the difference of the logs.

Special Case

lo g_{a} (\frac{1}{x}) = lo g_{a} 1 - lo g_{a} x = 0 - lo g_{a} x = - lo g_{a} x

Remember

Division INSIDE the log becomes subtraction OUTSIDE.

Numerator comes first (positive), denominator is subtracted.

Worked Example02

Expand $ln (\frac{x ^{2}}{y ^{3}})$ fully.

•

1. Apply division law:

ln (\frac{x ^{2}}{y ^{3}}) = ln (x^{2}) - ln (y^{3})

•

2. Apply power law to each term:

•

ln (x^{2}) = 2 ln x

and

ln (y^{3}) = 3 ln y

Answer: $2 ln x - 3 ln y$

Law 3: Power Law

Law 3

lo g_{a} (x^{n}) = n lo g_{a} x

In words: The log of a power equals the power times the log.

Works for any n

Fractions:

lo g_{a} (x) = lo g_{a} (x^{1/2}) = \frac{1}{2} lo g_{a} x

Negatives:

lo g_{a} (x^{- 2}) = - 2 lo g_{a} x

Remember

The power comes DOWN from inside the log to become a multiplier outside.

Worked Example03

Write $2 lo g_{3} x + lo g_{3} 5 - lo g_{3} y$ as a single logarithm.

•

1. Use power law on the first term:

2 lo g_{3} x = lo g_{3} (x^{2})

•

2. Now we have:

lo g_{3} (x^{2}) + lo g_{3} 5 - lo g_{3} y

•

3. Combine using multiplication law:

lo g_{3} (x^{2}) + lo g_{3} 5 = lo g_{3} (5 x^{2})

•

4. Combine using division law:

lo g_{3} (5 x^{2}) - lo g_{3} y = lo g_{3} (\frac{5 x ^{2}}{y})

Answer: $lo g_{3} (\frac{5 x ^{2}}{y})$

Combining the Laws

Complex expressions often require multiple laws. Here's the strategy:

Expanding

Work from outside in:

Split fractions and products (division/multiplication laws)

Bring down powers (power law)

Condensing

Reverse order:

Turn coefficients into powers (power law)

Combine additions/subtractions into products/quotients

Same Base Required

All terms must have the same base to combine them!

Worked Example04

Simplify $lo g_{4} 8 + lo g_{4} 2$ without a calculator.

•

1. Use multiplication law:

lo g_{4} 8 + lo g_{4} 2 = lo g_{4} (8 \times 2)

•

2. Calculate the product:

= lo g_{4} 16

•

3. Evaluate: What power of 4 gives 16?

•

4^{2} = 16

, so

lo g_{4} 16 = 2

Answer: 2

Change of Base Formula

Formula

lo g_{a} b = \frac{l o g _{c} b}{l o g _{c} a}

This lets you convert a logarithm from one base to another. Most commonly used with natural logs or common logs for calculator computation:

Using ln

lo g_{a} b = \frac{l n b}{l n a}

Using log₁₀

lo g_{a} b = \frac{l o g b}{l o g a}

When to Use

Use change of base when you need to calculate a logarithm with a base your calculator doesn't have directly.

Worked Example05

Use change of base to evaluate $lo g_{5} 20$ to 3 significant figures.

•

1. Apply change of base formula using ln:

•

lo g_{5} 20 = \frac{l n 20}{l n 5}

•

2. Calculate using calculator:

•

ln 20 = 2.996...

and

ln 5 = 1.609...

•

3. Divide:

\frac{2.996}{1.609} = 1.861...

Answer: 1.86 (3 s.f.)

Worked Example06

Given that $lo g_{2} 3 = 1.585$ , find the value of $lo g_{2} 12$ .

•

1. Express 12 in terms of factors involving 2 and 3:

•

12 = 4 \times 3 = 2^{2} \times 3

•

2. Apply multiplication law:

lo g_{2} 12 = lo g_{2} (2^{2} \times 3) = lo g_{2} (2^{2}) + lo g_{2} 3

•

3. Simplify:

lo g_{2} (2^{2}) = 2

and

lo g_{2} 3 = 1.585

•

4. Add:

2 + 1.585 = 3.585

Answer: 3.585

Common Errors

Critical Warning

There is NO addition or subtraction law!

WRONGWHY IT'S WRONG

lo g (x + y) = lo g x + lo g y

Addition inside ≠ addition outside. Only multiplication becomes addition.

lo g (x - y) = lo g x - lo g y

Subtraction inside ≠ subtraction outside. Only division becomes subtraction.

(lo g x)^{n} = n lo g x

The power must be ON the argument:

lo g (x^{n}) = n lo g x

\frac{l o g x}{l o g y} = lo g (\frac{x}{y})

Division OF logs ≠ log of quotient

Remember

lo g (x + y)

and

lo g (x - y)

cannot be simplified using the laws!

Worked Example07

Expand $ln (\frac{x y}{z ^{2}})$ fully.

•

1. Apply division law:

ln (\frac{x y}{z ^{2}}) = ln (x y) - ln (z^{2})

•

2. Apply multiplication law to numerator:

ln (x y) = ln x + ln (y)

•

3. Apply power law:

ln (y) = ln (y^{1/2}) = \frac{1}{2} ln y

•

4. Apply power law:

ln (z^{2}) = 2 ln z

•

5. Combine:

ln x + \frac{1}{2} ln y - 2 ln z

Answer: $ln x + \frac{1}{2} ln y - 2 ln z$

Worked Example08

Solve $lo g_{2} x + lo g_{2} (x - 2) = 3$

•

1. Use multiplication law to combine:

lo g_{2} [x (x - 2)] = 3

•

2. Convert to index form:

x (x - 2) = 2^{3} = 8

•

3. Expand and rearrange:

x^{2} - 2 x = 8

→

x^{2} - 2 x - 8 = 0

•

4. Factorise:

(x - 4) (x + 2) = 0

•

5. Solutions:

x = 4

or

x = - 2

•

6. Check validity: Need

x > 0

AND

x - 2 > 0

, so

x > 2

. Only

x = 4

is valid.

Answer: $x = 4$

Practice

Question 1: Expand

lo g_{3} (9 x^{2})

fully.

•

1. Apply multiplication law:

lo g_{3} (9 x^{2}) = lo g_{3} 9 + lo g_{3} (x^{2})

•

2. Apply power law:

lo g_{3} (x^{2}) = 2 lo g_{3} x

•

3. Evaluate:

lo g_{3} 9 = 2

since

3^{2} = 9

Answer: $2 + 2 lo g_{3} x$

Practice

Question 2: Solve

lo g_{5} x + lo g_{5} (x + 4) = 1

•

1. Combine using multiplication law:

lo g_{5} [x (x + 4)] = 1

•

2. Convert to index form:

x (x + 4) = 5^{1} = 5

•

3. Expand and rearrange:

x^{2} + 4 x - 5 = 0

•

4. Factorise:

(x + 5) (x - 1) = 0

, so

x = - 5

or

x = 1

•

5. Check validity: Need

x > 0

AND

x + 4 > 0

. Only

x = 1

is valid.

Answer: $x = 1$

Practice

Question 3: Use change of base to find

lo g_{7} 50

to 3 significant figures.

•

1. Apply change of base:

lo g_{7} 50 = \frac{l n 50}{l n 7}

•

2. Calculate:

ln 50 = 3.912...

and

ln 7 = 1.946...

•

3. Divide:

\frac{3.912}{1.946} = 2.010...

Answer: 2.01 (3 s.f.)

Summary

The Three Laws

LawFormulaIn Words

Multiplication

lo g_{a} (x y) = lo g_{a} x + lo g_{a} y

Log of product = sum of logs

Division

lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y

Log of quotient = difference of logs

Power

lo g_{a} (x^{n}) = n lo g_{a} x

Log of power = power × log

Change of Base

lo g_{a} b = \frac{l n b}{l n a} = \frac{l o g b}{l o g a}

Never Forget

lo g (x + y) \neq = lo g x + lo g y

Must have same base to combine!