Exponential and Logarithmic Functions

What is the general form of an exponential function?

f(x) = a × b^x, where a and b are real numbers and b is positive.

How does the behavior of an exponential function change based on the value of b?

If 0 < b < 1, the function decays (decreases as x increases). If b > 1, the function grows (increases as x increases).

What is the domain and range of a standard exponential function?

The domain is (-∞, ∞) and the range is (0, ∞).

Explain what it means for a function to be one-to-one.

A function is one-to-one if each x-value corresponds to a unique y-value, allowing an inverse function to exist.

How can you solve an equation where the bases are the same, like 2^{3x} = 2^{4x-1}?

You equate the exponents: 3x = 4x - 1, then solve for x to get x = 1.

State the laws of exponents for multiplication and division.

b^x × b^y = b^{x+y} and b^x / b^y = b^{x-y}.

Why is it incorrect to assume (x-y)^2 equals x^2 - y^2?

The correct expansion is x^2 - 2xy + y^2.

What conditions must a function meet to have an inverse?

The function must be one-to-one and onto (surjective).

Describe the horizontal line test.

A graphical method to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, it's not one-to-one.

How is the inverse of a function found algebraically?

You swap x and y in the equation and solve for y.

Define a logarithmic function and its relationship with an exponential function.

A logarithmic function is the inverse of an exponential function: y = b^x if and only if log_b(y) = x.

What are the properties of natural logarithms (ln)?

The base is e, approximately 2.718, and ln(x) is the inverse of e^x.

What is the domain and range of a logarithmic function?

The domain is (0, ∞) and the range is (-∞, ∞).

Explain the laws of logarithms for multiplication, exponentiation, and division.

log_b(xy) = log_b(x) + log_b(y), log_b(x^r) = rlog_b(x), log_b(x/y) = log_b(x) - log_b(y).

How can you find the inverse of a function graphically?

By reflecting the function's graph across the line y = x.