Eight Rules from Exponent Laws
Each logarithm property follows from a corresponding exponent law, so every formula below can be derived from that law.
All the properties below use these conditions:
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Whenever or is used as a logarithm base, that base must also be positive and not equal to one.
| Rule | Identity |
|---|---|
| Logarithm of the base | |
| Logarithm of one | |
| Inverse relationship | |
| Product | |
| Quotient | |
| Power | |
| Change of base | |
| Chain |
The first three rules come directly from , , and the inverse relationship between logarithms and exponents.
Every logarithm rule follows from the matching exponent rule. A logarithm reports the exponent that produces a given number, so a rule about exponents becomes a rule about logarithms.
Let:
Multiplying and adds their exponents:
Taking the base- logarithm gives:
Using the same substitutions, division subtracts the exponents:
The calculation gives:
Let , so . For any real , the condition makes well-defined:
This proof covers real exponents directly. Repeated multiplication would explain only positive integer exponents.
Let , so . Take the logarithm of both sides using any valid base :
Because , this proves the change-of-base formula. Choosing gives the reciprocal relationship:
After both logarithms are written with the same valid base, appears once in the numerator and once in the denominator, so the two factors cancel:
Calculate .
Because , the inverse relationship and power rule give:
The result also answers the original exponent question: .
The four problems below practice the product, quotient, power, and change-of-base rules together. Work each one before reading the worked solutions, then compare your steps with the answers.
Simplify three expressions that each need one rule: , , and .
If and , express in terms of and .
A city has residents and is modeled as growing by at the end of each year. Under this fixed-rate model, after how many complete years will its population first reach at least ?
Dini deposits in an account paying interest compounded annually. Assume no further deposits, withdrawals, fees, or rate changes. After how many complete years will the balance first reach at least ?
For a numerical logarithm, identify the matching power. For a growth threshold, solve for time and check the neighboring whole years against the original model. Keep every logarithm argument positive when combining or expanding logarithms.
Problem 1. Since :
Use the quotient rule:
Since :
Problem 2. Use base for the numerator and denominator:
The reciprocal rule gives . Therefore:
Problem 3. To find when the population reaches its target, use the fixed-rate model:
Solve the continuous equation first:
The model updates after complete years, so check the neighboring integers:
The population first reaches the target after . This result assumes that the stated growth rate stays fixed for all years. A changing growth rate would give a different answer.
Problem 4. To find when the balance reaches its target, use the annual-compounding model:
Solve the continuous equation:
Interest is credited only after each complete year, so compare years and :
The balance first reaches the target after .
Published: . Updated: .