Properties of Logarithms - Laws, Formulas, Proofs
Properties of Logarithms - Laws, Formulas, Proofs
TL;DR
The properties of logarithms are seven identities that simplify logarithmic expressions: the product rule (( \log_b(xy) = \log_b x + \log_b y )), quotient rule (( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y )), power rule (( \log_b(x^n) = n \log_b x )), change-of-base formula, and three special values (( \log_b 1 = 0 ), ( \log_b b = 1 ), ( \log_b b^n = n )).
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Madhukar V Last updated on May 15, 2022 9 min read
What Is a Logarithm? (Quick Refresher)
A logarithm answers the question "to what power must I raise the base to get this number?" If ( b^y = x ), then ( \log_b(x) = y ). Example: ( \log_2(8) = 3 ) because ( 2^3 = 8 ).
The two most common bases are:
- Common logarithm — base 10, often written as ( \log(x) ) with no subscript
- Natural logarithm — base ( e \approx 2.71828 ), written ( \ln(x) )
Every property below holds for any base ( b ) (with ( b > 0 ) and ( b \neq 1 )).
What Are the Properties of Logarithms?
The seven core properties, in compact form:
| Property | Identity | When You Use It |
|---|---|---|
| Product | ( \log_b(xy) = \log_b x + \log_b y ) | Multiplication → addition |
| Quotient | ( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y ) | Division → subtraction |
| Power | ( \log_b(x^n) = n \log_b x ) | Exponent → coefficient |
| Change of base | ( \log_b x = \frac{\log_c x}{\log_c b} ) | Converting between bases |
| Log of 1 | ( \log_b 1 = 0 ) | Anything to the 0 is 1 |
| Log of base | ( \log_b b = 1 ) | Anything to the 1 is itself |
| Log of base power | ( \log_b(b^n) = n ) | Inverse of exponential |
How Do the Three Main Properties Work? (With Proofs)
Product Rule
( \log_b(xy) = \log_b x + \log_b y )
Proof. Let ( m = \log_b x ) and ( n = \log_b y ). Then ( x = b^m ) and ( y = b^n ), so:
[ xy = b^m \cdot b^n = b^{m+n} ]
Taking ( \log_b ) of both sides: ( \log_b(xy) = m + n = \log_b x + \log_b y ). ∎
Example. ( \log_2(8 \times 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5 ).
Quotient Rule
( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y )
Proof. Same setup as above: ( \frac{x}{y} = \frac{b^m}{b^n} = b^{m-n} ), so ( \log_b\left(\frac{x}{y}\right) = m - n ). ∎
Example. ( \log_3\left(\frac{81}{9}\right) = \log_3 81 - \log_3 9 = 4 - 2 = 2 ).
Power Rule
( \log_b(x^n) = n \log_b x )
Proof. Let ( m = \log_b x ), so ( x = b^m ). Then ( x^n = (b^m)^n = b^{mn} ), giving ( \log_b(x^n) = mn = n \log_b x ). ∎
Example. ( \log_2(8^3) = 3 \log_2 8 = 3 \times 3 = 9 ).
How Do You Use the Change-of-Base Formula?
Calculators only compute ( \log ) (base 10) and ( \ln ) (base e). For any other base, use the change-of-base formula:
[ \log_b x = \frac{\log_c x}{\log_c b} ]
where ( c ) is any base your calculator supports.
Example. Compute ( \log_5(125) ) using base-10 logs.
[ \log_5(125) = \frac{\log 125}{\log 5} \approx 3 ]
Check: ( 5^3 = 125 ) ✓.
Special case — change to natural log:
[ \log_b x = \frac{\ln x}{\ln b} ]
What Are the Three Special-Value Properties?
These follow directly from the definition of a logarithm.
Log of 1 = 0
( \log_b 1 = 0 ) (any base ( b > 0 ), ( b \neq 1 )).
Because ( b^0 = 1 ) for any non-zero base ( b ).
Log of the Base Itself = 1
( \log_b b = 1 )
Because ( b^1 = b ).
Log of a Power of the Base = the Exponent
( \log_b(b^n) = n )
This is the inverse-function property.
What Are the Properties of the Natural Logarithm (ln)?
The natural logarithm ( \ln(x) ) is just ( \log_e(x) ). Every property above carries over verbatim with ( e ); we restate them here because ( \ln ) is the form you'll meet most often in calculus, physics, and finance.
| Property | Identity for ( \ln ) |
|---|---|
| Product | ( \ln(xy) = \ln x + \ln y ) |
| Quotient | ( \ln\left(\frac{x}{y}\right) = \ln x - \ln y ) |
| Power | ( \ln(x^n) = n \ln x ) |
| Change of base | ( \ln x = \frac{\log x}{\log e} ) |
| ln of 1 | ( \ln 1 = 0 ) |
| ln of e | ( \ln e = 1 ) |
| ln of e^n | ( \ln(e^n) = n ) |
| Exponential inverse | ( e^{\ln x} = x ) for all ( x > 0 ) |
Calculus identity (bonus). The natural log is the unique logarithm whose derivative is ( \frac{1}{x} ):
[ \frac{d}{dx}\ln x = \frac{1}{x}, \quad x > 0 ]
Why Were Logarithms Invented? (The Real-World GROUND)
"Seeing there is nothing… that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications…" — John Napier, 1614.
Logarithms exist because of a specific 17th-century problem: astronomers and navigators needed to multiply enormous numbers, and multiplication by hand took hours. John Napier published Mirifici Logarithmorum Canonis Descriptio in 1614. His logarithm tables converted multiplication into addition.
Today, logarithmic scales describe enormous ranges of natural phenomena:
- The Richter scale — magnitude-7 earthquake releases 10× the energy of magnitude-6.
- The decibel scale — sound intensity doubles roughly every 3 dB.
- The pH scale — pH 3 is 10× more acidic than pH 4.
A Worked Example
Simplify ( \log_2(8 \cdot 4) - \log_2 4 ).
The intuitive (wrong) approach. A student applies the product rule incorrectly:
( \log_2(8 \cdot 4) = \log_2 8 \cdot \log_2 4 = 3 \cdot 2 = 6 ).
Then ( 6 - \log_2 4 = 6 - 2 = 4 ).
The correct method.
( \log_2(8 \cdot 4) - \log_2 4 = \log_2 8 = 3 ).
What Are the Most Common Mistakes With Properties of Logarithms?
Mistake 1: Splitting ( \log(x+y) ) as ( \log x + \log y )
Where it slips in: Trying to "distribute" the log over an addition.
Don't do this: ( \log(x+y) = \log x + \log y ). This is false.
Mistake 2: Confusing ( \log(x^n) ) with ( (\log x)^n )
Where it slips in: ( \log(x^2) ) and ( (\log x)^2 ) look similar but mean different things.
Don't do this: Treating ( (\log x)^2 ) and ( \log(x^2) ) as equal.
Mistake 3: Treating ( \log 0 ) as 0
Where it slips in: Recalling that ( \log 1 = 0 ).
Don't do this: Writing ( \log 0 = 0 ).
The Mathematicians Who Shaped Logarithms
John Napier (1550–1617, Scotland) — Invented logarithms in 1614.
Henry Briggs (1561–1630, England) — Developed the base-10 logarithm tables.
Leonhard Euler (1707–1783, Switzerland) — Established the connection between logarithms and the number ( e ).
A Practical Next Step
- Simplify ( \log 100 + \log 10 ) using the product rule.
- Simplify ( \log_2(64) - \log_2(8) ) using the quotient rule.
- Compute ( \log_3(81) ) — without a calculator.
Frequently Asked Questions
What are the 7 properties of logarithms?
The seven core properties: product, quotient, power, change of base, log of 1, log of base, and log of base power.
What is the change-of-base formula?
( \log_b x = \frac{\log_c x}{\log_c b} ) for any valid base ( c ).
Is ( \log(x+y) = \log x + \log y )?
No. This is the most common log mistake.
Why is ( \log_b 1 = 0 )?
Because ( b^0 = 1 ) for any non-zero base ( b ).
What is the difference between ( \log ) and ( \ln )?
( \log ) usually means base-10 logarithm, and ( \ln ) means base-e logarithm.
How are the properties of logarithms used in real life?
Logarithmic scales describe earthquake magnitude, sound intensity, acid concentration, and more.