Glossary
Change of Base Formula
A formula that allows logarithms to be converted from one base to another, typically to base 10 or base e for calculator use: `log_b(M) = log_c(M) / log_c(b)`.
Example:
To evaluate log_3(10) on a calculator, you can use the Change of Base Formula to write it as ln(10) / ln(3).
Equations
Mathematical statements asserting that two expressions are equal, often involving variables that need to be solved for.
Example:
Solving the equation 3^(x+1) = 81 requires recognizing that 81 is 3 to the power of 4.
Exponential Functions
Functions where the variable appears in the exponent, typically in the form `f(x) = ab^(x-h) + k`, modeling growth or decay.
Example:
The spread of a virus can often be modeled by an exponential function, showing rapid initial increase.
Extraneous Solutions
Solutions that arise during the algebraic solving process but are not valid for the original equation, often due to domain restrictions (e.g., logarithms only defined for positive arguments).
Example:
When solving log(x-2) + log(x) = log(3), you might get x = -1 as a potential answer, but it's an extraneous solution because log(-1) is undefined.
Inequalities
Mathematical statements comparing two expressions using symbols like <, >, ≤, or ≥, indicating that one expression is greater than or less than the other.
Example:
To find when a population grows beyond a certain limit, you might solve an exponential inequality like 100 * (1.05)^t > 200.
Inverse Relationship
The fundamental connection between exponential and logarithmic functions, where one 'undoes' the other, allowing conversion between forms to solve equations.
Example:
Because of their inverse relationship, if e^x = 7, then ln(7) = x.
Inverse of a Function
A function that reverses the action of the original function, meaning if `f(a) = b`, then `f^-1(b) = a`. It is found by swapping x and y and solving for y.
Example:
If f(x) = 2x + 1, its inverse of a function f^-1(x) = (x-1)/2 will return the original input.
Logarithmic Functions
Functions that are the inverse of exponential functions, typically in the form `f(x) = a log_b (x - h) + k`, used to solve for exponents or model phenomena with decreasing rates of change.
Example:
The Richter scale, which measures earthquake intensity, is based on a logarithmic function.
Power Rule (Logarithms)
A property stating that the logarithm of a number raised to a power is the product of the power and the logarithm of the number: `log_b(M^p) = p * log_b(M)`.
Example:
With the Power Rule, log(x^3) can be rewritten as 3 log(x), making it easier to solve for x.
Product Rule (Logarithms)
A property stating that the logarithm of a product is the sum of the logarithms of the individual factors: `log_b(MN) = log_b(M) + log_b(N)`.
Example:
Using the Product Rule, log_5(25x) can be expanded to log_5(25) + log_5(x), which simplifies to 2 + log_5(x).
Properties of Exponents
Rules that govern how exponents behave in mathematical operations, such as multiplying powers with the same base or raising a power to another power.
Example:
Using the properties of exponents, (x^2)^3 simplifies to x^6, not x^5.
Properties of Logarithms
Rules that simplify logarithmic expressions, including the product, quotient, and power rules, which are derived from the properties of exponents.
Example:
The properties of logarithms allow us to combine log(x) + log(y) into a single term, log(xy), which is useful for solving equations.
Quotient Rule (Logarithms)
A property stating that the logarithm of a quotient is the difference of the logarithms of the numerator and denominator: `log_b(M/N) = log_b(M) - log_b(N)`.
Example:
The Quotient Rule helps simplify log_2(x/8) into log_2(x) - log_2(8), which is log_2(x) - 3.