Exponents & Logarithms

Question 1

Characteristics of Exponential Graphs

Answer 1

Shape – Exponential graphs have a smooth, continuous curve that increases or decreases rapidly.

Growth or Decay – If the base a is greater than 1, the function shows exponential growth. If 0 < a < 1, the function shows exponential decay.

Y-Intercept – The graph always passes through the point (0,1) because a^0 = 1 for any positive base a.

Asymptote – The graph has a horizontal asymptote at y = 0, meaning it approaches but never touches the x-axis.

Domain and Range – The domain is all real numbers, while the range is y > 0.

Increasing or Decreasing – The function is always increasing if a > 1 and always decreasing if 0 < a < 1.

No Symmetry – Exponential graphs are not symmetric about the y-axis or origin.

Question 2

True or False

a = bx and log b a = x are equivalent statements

Answer 2

True

Question 3

What is a logarithmic statement?

Answer 3

A logarithm statement is a mathematical equation that expresses the relationship between exponents and logarithms. It is written in the form:

log_b(x) = y

This means that b raised to the power of y equals x. In other words, it is the inverse of an exponential function:

b^y = x ⇔ log_b(x) = y

For example:

log₂(8) = 3 because 2³ = 8

This statement shows that the logarithm finds the exponent (y) needed to reach a given number (x) using the base b.

Question 4

Define the term asymptote

Answer 4

An asymptote is a line that a graph gets closer and closer to but never actually touches. It shows the behavior of a function as it moves towards infinity or a certain point.