Math
Master the quantitative skills used in science and healthcare: significant figures, unit conversions, proportions, functions, exponents, logarithms, statistics, and introductory calculus.
Exponents & Logarithms
Sign in to track progressExponents and logarithms are essential for describing rapid growth, decay, and pH scales in chemistry and biology.
Exponential Functions An exponential function has the form f(x) = a·bˣ where b > 0 and b ≠ 1. The variable is in the exponent.
Exponential Equations When bases are equal, set the exponents equal:
2ˣ = 2³ → x = 3
Solving Exponential Expressions Use exponent rules:
xᵃ · xᵇ = xᵃ⁺ᵇ
xᵃ / xᵇ = xᵃ⁻ᵇ
(xᵃ)ᵇ = xᵃᵇ
x⁻ᵃ = 1/xᵃ
x⁰ = 1 (x ≠ 0)
Logarithms A logarithm is the inverse of an exponent. log_b(x) asks: "To what power must I raise b to get x?"
Common logarithms use base 10: log(x) means log₁₀(x).
Inverse Relationship Exponential and logarithmic functions undo each other:
log_b(bˣ) = x
b raised to the power log_b(x) = x
Solving Logarithmic Equations Rewrite in exponential form: log₁₀(x) = 2 → 10² = x → x = 100
Graphs The graphs of y = bˣ and y = log_b(x) are reflections of each other across the line y = x.
Key Points
- Exponents follow product, quotient, power, and negative-exponent rules
- Logarithms are the inverse operation of exponents
- Common logarithms use base 10
- log_b(x) = y means bʸ = x
- Exponential and logarithmic graphs are reflections across y = x