Basic Properties of Logarithms

Four properties, each one falling straight out of the definition: log_b(1) = 0, log_b(b) = 1, log_b(b^x) = x, and b^(log_b x) = x — showing that logarithms and exponentials undo each other.

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Four properties do most of the work in logarithms, and none of them need to be memorised blindly. Each one falls straight out of the definition: a logarithm asks what power the base must be raised to in order to give a number. Keep that question in mind and every rule below explains itself.

Concept A quick reminder

Take \log_4(16) . The question is: 4 raised to what power gives 16?

4^2 = 16 , so the exponent is 2.
⟹ log₄(16) = 2

Throughout what follows the base b must satisfy b > 0 and b \neq 1 , the same conditions that make any logarithm defined.

Property 1 The logarithm of 1

\log_b(1) = 0

Why: any number raised to the power zero equals 1, so b^0 = 1 . The exponent needed to reach 1 is therefore always 0 — whatever the base happens to be.

Property 2 The logarithm of the base itself

\log_b(b) = 1

Why: because b^1 = b . Reaching the base requires exactly one copy of it.

\log_2(2) = 1
\log_5(5) = 1
\log_{10}(10) = 1
Property 3 The logarithm undoes the exponent

\log_b(b^x) = x

Why: b^x is precisely the number you get by raising the base to the power x . Asking which exponent produces it hands back x directly.

\log_2(2^3) = 3
\log_3(3^5) = 5
\log_{10}(10^7) = 7
Property 4 The exponent undoes the logarithm

b^{\log_b(x)} = x

Why: the logarithm works out the required exponent, and raising the base to that exponent returns the original number. This holds provided x > 0 .

2^{\log_2 4} = 2^2 = 4
3^{\log_3 27} = 3^3 = 27
10^{\log_{10} 50} = 50

Properties 3 and 4 together say something important: the logarithm and the exponential are inverse operations. Applied one after the other, in either order, they cancel out and leave you where you started.

Note The four properties together
x x raise, then take log x x take log, then raise
\log_b(1) = 0
\log_b(b) = 1
\log_b(b^x) = x
b^{\log_b(x)} = x
Example Simplifying without a calculator

Simplify \log_7(1) + \log_7(7) + \log_7(7^4) .

By Property 1, \log_7(1) = 0 .
By Property 2, \log_7(7) = 1 .
By Property 3, \log_7(7^4) = 4 .
Add the three results: 0 + 1 + 4 .
⟹ 5
Example Cancelling in both directions

Evaluate 5^{\log_5 12} and \log_6(6^9) .

The first has matching base 5 inside and outside, so Property 4 applies directly.
5^{\log_5 12} = 12 , with no need to find the logarithm itself.
The second matches Property 3, with base 6 throughout.
\log_6(6^9) = 9 .
⟹ 12 and 9

Both cancellations require the bases to match. If they differ, neither property applies and the expression cannot be simplified this way.

Note Mistakes to avoid
Writing \log_b(1) = 1 — it is 0, because b^0 = 1 .
Writing \log_b(b) = 0 — it is 1, because b^1 = b .
Cancelling when the bases do not match, such as in \log_2(3^5) .
Applying b^{\log_b(x)} = x when x is zero or negative; the property needs x > 0 .
Forgetting the base conditions b > 0 and b \neq 1 that every property depends on.
Summary
  1. log_b(1) = 0, because any base raised to the power 0 gives 1.
  2. log_b(b) = 1, because b¹ = b.
  3. log_b(bˣ) = x — the logarithm hands back the exponent.
  4. b^(log_b(x)) = x for x > 0 — raising the base undoes the logarithm.
  5. Properties 3 and 4 show that logarithms and exponentials are inverse operations, but only when the bases match.
231. Arithmetic Sequences and Series
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