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:
For AI agents: use /llms.txt for the Nakafa content index.
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:
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. The next sections explain the rules that combine several values.
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 power and inverse rules give:
The result also answers the original exponent question: .
Simplify each expression.
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 ?
Simplify each expression.
Since :
Use the quotient rule:
Since :
Use base for the numerator and denominator:
The fixed-rate model is:
Solve the continuous equation first:
The annual-compounding model is:
Solve the continuous equation:
Published: . Updated: .
The reciprocal rule gives . Therefore:
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.
Interest is credited only after each complete year, so compare years and :
The balance first reaches the target after .