Properties of Logarithms

Learn the basic properties of logarithms: the log of 1, the log of the base, the two inverse properties and the one-to-one property.

Advanced25 min18 problems
Practice Problems

Definition

The basic properties of logarithms all come from one fact: a logarithm is an exponent. For a base b>0b > 0 with b≠1b \neq 1:
log⁡bx=y  ⟺  by=x\log_b x = y \iff b^y = x
The five basic properties
  1. 1.Log of 1: log⁡b1=0\log_b 1 = 0, because b0=1b^0 = 1
  2. 2.Log of the base: log⁡bb=1\log_b b = 1, because b1=bb^1 = b
  3. 3.Inverse property (log of a power of the base): log⁡b(bx)=x\log_b(b^x) = x for every real number xx
  4. 4.Inverse property (base raised to a log): blog⁡bx=xb^{\log_b x} = x for every x>0x > 0
  5. 5.One-to-one property: if log⁡bM=log⁡bN\log_b M = \log_b N (with M>0M > 0, N>0N > 0), then M=NM = N
What is not allowed: the argument of a logarithm must be positive. log⁡b0\log_b 0 and log⁡b(−5)\log_b(-5) are undefined, because byb^y is positive for every exponent yy.
Properties 3 and 4 say that y=log⁡bxy = \log_b x and y=bxy = b^x are inverse functions: one undoes the other.
Overview: the three rules for combining logarithms (each one has its own lesson)
RuleFormulaLesson
Product rulelog⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b NProduct Rule for Logarithms
Quotient rulelog⁡bMN=log⁡bM−log⁡bN\log_b \frac{M}{N} = \log_b M - \log_b NQuotient Rule for Logarithms
Power rulelog⁡b(Mp)=plog⁡bM\log_b(M^p) = p \log_b MPower Rule for Logarithms
In this lesson we practice the five basic properties. The three rules in the table are taught step by step in their own lessons.

Try it now

What is log⁡61\log_6 1?

Worked Examples

Evaluate log⁡91+log⁡99\log_9 1 + \log_9 9.
1

Evaluate log⁡91\log_9 1

Which power of 99 gives 11? 90=19^0 = 1 → log⁡91=0\log_9 1 = 0
2

Evaluate log⁡99\log_9 9

Which power of 99 gives 99? 91=99^1 = 9 → log⁡99=1\log_9 9 = 1
3

Add the results

0+10 + 1

Common Mistakes

Writing log⁡b1=1\log_b 1 = 1 or log⁡b1=b\log_b 1 = b

Why it's wrong: The logarithm is the exponent, not the result. The question is: which power of bb gives 11?

Correct: b0=1b^0 = 1, so log⁡b1=0\log_b 1 = 0 for every allowed base. For example log⁡71=0\log_7 1 = 0.

Thinking log⁡b0=0\log_b 0 = 0 or that the log of a negative number is a negative number

Why it's wrong: A positive base raised to any power is always positive, so no exponent gives 00 or a negative number.

Correct: log⁡b0\log_b 0 and log⁡b(−4)\log_b(-4) are undefined. The argument must be greater than 00, while the value of a logarithm can be negative: log⁡218=−3\log_2 \frac{1}{8} = -3.

Cancelling a power and a logarithm with different bases: 2log⁡39=92^{\log_3 9} = 9

Why it's wrong: blog⁡bx=xb^{\log_b x} = x only works when the base of the power and the base of the logarithm are the same number.

Correct: Here the bases differ: log⁡39=2\log_3 9 = 2, so 2log⁡39=22=42^{\log_3 9} = 2^2 = 4, not 99.

Using the one-to-one property and not checking the arguments

Why it's wrong: Setting the arguments equal can produce a value that makes an argument zero or negative, where the original logarithm does not exist.

Correct: Always put each candidate back into every argument. Keep it only if all arguments are positive.

Interactive Visual

Function graph

024681012-4-2024xy(1, 0)(2, 1)x = 0

Tap or hover over the graph to read off values.

y = log_b(x)y = 1Asymptote

Table of values

x0.5124810
log_b(x)-101233.32193
1111111

Interactive Sandbox

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Practice Problems

18 problems
Problem 1 of 18
Easy
What is log⁡61\log_6 1?

Why It Matters

The basic properties are the tools you use every time you work with logarithms:
  • Evaluating without a calculator: log⁡264=log⁡2(26)=6\log_2 64 = \log_2(2^6) = 6 in one line
  • Solving equations: the inverse properties remove a logarithm or an exponential, so log⁡3x=4\log_3 x = 4 becomes x=34=81x = 3^4 = 81
  • Comparing: the one-to-one property turns log⁡5(2x+1)=log⁡511\log_5(2x + 1) = \log_5 11 into the simple equation 2x+1=112x + 1 = 11
  • Science scales: decibels, pH and the Richter scale all use log⁡\log. A sound at the hearing threshold has 10log⁡1=010 \log 1 = 0 decibels
  • Avoiding errors: knowing that log⁡b0\log_b 0 and logs of negative numbers are undefined tells you which solutions to throw away
The product, quotient and power rules (and the change of base formula) are all built on these five properties.

Real World Applications

Sound Level in Decibels

The loudness of a sound is $L = 10 \log \frac{I}{I_0}$ decibels, where $I_0$ is the quietest sound a person can hear. The basic properties make the calculation quick.

Example:

At the hearing threshold I=I0I = I_0, so L=10log⁡1=10⋅0=0L = 10 \log 1 = 10 \cdot 0 = 0 decibels.

1Try It Yourself

A rock concert has an intensity of I=1011⋅I0I = 10^{11} \cdot I_0, so II0=1011\frac{I}{I_0} = 10^{11}.

What is the sound level of the concert in decibels?

Step 1: Write the mathematical expression

Compute L=10log⁡(1011)L = 10 \log(10^{11}):

Acidity and pH

Chemists measure acidity with $\text{pH} = -\log[H^+]$, where $[H^+]$ is the hydrogen ion concentration in moles per liter. The inverse property turns a pH back into a concentration.

Example:

Pure water has [H+]=10−7[H^+] = 10^{-7}, so pH=−log⁡(10−7)=−(−7)=7\text{pH} = -\log(10^{-7}) = -(-7) = 7.

2Try It Yourself

Lemon juice has a pH of 22, so −log⁡[H+]=2-\log[H^+] = 2 and log⁡[H+]=−2\log[H^+] = -2.

What is the hydrogen ion concentration [H+][H^+] of lemon juice?

Step 1: Write the mathematical expression

Use 10log⁡x=x10^{\log x} = x and compute [H+]=10−2[H^+] = 10^{-2}:

Rounds in a Knockout Tournament

In a knockout tournament half of the teams drop out in every round. With $2^n$ teams there are $\log_2(2^n) = n$ rounds until one winner is left.

Example:

With 16=2416 = 2^4 teams there are log⁡216=log⁡2(24)=4\log_2 16 = \log_2(2^4) = 4 rounds.

3Try It Yourself

A school chess tournament starts with 128128 players. Every game has a winner, and the loser drops out.

How many rounds are needed to find the champion?

Step 1: Write the mathematical expression

Compute log⁡2128\log_2 128:

Key Takeaways

  • 1log⁡b1=0\log_b 1 = 0 because b0=1b^0 = 1, and log⁡bb=1\log_b b = 1 because b1=bb^1 = b
  • 2Inverse property: log⁡b(bx)=x\log_b(b^x) = x for every real xx
  • 3Inverse property: blog⁡bx=xb^{\log_b x} = x for every x>0x > 0, but only when both bases are the same
  • 4One-to-one property: if log⁡bM=log⁡bN\log_b M = \log_b N, then M=NM = N; always check that the arguments are positive
  • 5log⁡b0\log_b 0 and the logarithm of a negative number are undefined, while the value of a logarithm may be negative
  • 6The product, quotient and power rules build on these properties and are taught in their own lessons

Frequently Asked Questions

Any allowed base raised to the power 00 equals 11: 20=12^0 = 1, 100=110^0 = 1, e0=1e^0 = 1. The logarithm asks for that exponent, so the answer is always 00.
Any allowed base raised to the power 00 equals 11: 20=12^0 = 1, 100=110^0 = 1, e0=1e^0 = 1. The logarithm asks for that exponent, so the answer is always 00.
Yes. Only the argument has to be positive. For example log⁡100.01=−2\log_{10} 0.01 = -2 because 10−2=0.0110^{-2} = 0.01. The argument 0.010.01 is positive, the value −2-2 is negative.
In log⁡b(bx)\log_b(b^x) the argument bxb^x is always positive, so every real xx is allowed. In blog⁡bxb^{\log_b x} the logarithm log⁡bx\log_b x must exist first, and that needs x>0x > 0.
Each has its own lesson: Product Rule for Logarithms, Quotient Rule for Logarithms and Power Rule for Logarithms. They use the properties from this lesson, so it helps to master these first.

Glossary

Logarithm
The exponent yy with by=xb^y = x, written log⁡bx=y\log_b x = y
Base
The number bb in log⁡bx\log_b x; it must satisfy b>0b > 0 and b≠1b \neq 1
Argument
The number xx in log⁡bx\log_b x; it must be positive
Inverse properties
log⁡b(bx)=x\log_b(b^x) = x and blog⁡bx=xb^{\log_b x} = x: the logarithm and the exponential function with the same base undo each other
One-to-one property
If log⁡bM=log⁡bN\log_b M = \log_b N, then M=NM = N: different positive arguments never give the same logarithm
Inverse function
A function that reverses another one; y=log⁡bxy = \log_b x is the inverse function of y=bxy = b^x

Formula Card

Log of 1

log⁡b1=0\log_b 1 = 0

Because $b^0 = 1$

Log of the base

log⁡bb=1\log_b b = 1

Because $b^1 = b$

Inverse property 1

log⁡b(bx)=x\log_b(b^x) = x

For every real number $x$

Inverse property 2

blog⁡bx=xb^{\log_b x} = x

For every $x > 0$, same base on both levels

One-to-one property

log⁡bM=log⁡bN⇒M=N\log_b M = \log_b N \Rightarrow M = N

Check that $M > 0$ and $N > 0$

Domain

log⁡bx\log_b x exists only for x>0x > 0

$\log_b 0$ and logs of negative numbers are undefined

Today's challengeOne problem, three tries

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