Laws of Exponents
Fractional and Negative Exponents
What Is a Logarithm?
Logarithm Properties
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Change of Base Formula
Applications: Growth and Decay
Exponential growth and decay models are analyzed using derivatives to find rates of change, and integration techniques to find accumulated quantities.
Common Algebra Mistakes
Distributing exponents over addition
(a + b)² ≠ a² + b². The correct expansion is a² + 2ab + b². This is one of the most persistent algebra errors.
Confusing log rules with exponent rules
log(a + b) ≠ log(a) + log(b). The product rule is log(ab) = log(a) + log(b). The rules mirror exponent rules but apply to multiplication, not addition.
Practice Problems
Related Guides
Frequently Asked Questions
What is a logarithm?
A logarithm answers the question: "What exponent do I need?" Specifically, log_b(x) = y means b^y = x. For example, log₂(8) = 3 because 2³ = 8. Logarithms are the inverse of exponentiation.
What are the laws of exponents?
The key laws are: a^m × a^n = a^(m+n) for multiplying same-base powers, a^m / a^n = a^(m-n) for dividing, (a^m)^n = a^(mn) for power of a power, a^0 = 1 for any nonzero a, and a^(-n) = 1/a^n for negative exponents.
What is the change of base formula?
The change of base formula lets you evaluate any logarithm using a different base: log_b(x) = log_c(x) / log_c(b). This is especially useful for calculator computation, where you can convert to ln or log₁₀.
What does a fractional exponent mean?
A fractional exponent combines roots and powers: a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m). For example, 8^(2/3) = (³√8)² = 2² = 4. The denominator is the root, the numerator is the power.
What are real-world applications of exponential functions?
Exponential functions model population growth, radioactive decay, compound interest, bacterial reproduction, cooling/heating processes, and the spread of diseases. Any process with a constant percentage rate of change is exponential.
Why is ln (natural logarithm) special?
The natural logarithm (ln) uses base e ≈ 2.71828. It is special because the derivative of e^x is e^x (itself), making it fundamental in calculus. The number e arises naturally from continuous compound interest and appears throughout science and engineering.
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