IB Mathematics AA SL Exponential & logarithmic functions Study Notes
IB Mathematics AA SL Exponential & logarithmic functions Study Notes Offer a clear explanation of Exponential & logarithmic functions , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Exponential & logarithmic functions
Exponential and Logarithmic Function
Exponential Graphs
Exponential functions have the form \( f(x) = a^x \), where \( a \) is a positive real number, and \( a \neq 1 \).
Key Features:
- Growth and Decay: When \( a > 1 \), the function represents exponential growth. When \( 0 < a < 1 \), it represents exponential decay.
- Horizontal Asymptote: The graph has a horizontal asymptote at \( y = 0 \).
- y-intercept: The y-intercept is at \( (0, 1) \).
For example, the function \( f(x) = 2^x \) is an exponential growth function. The graph passes through \( (0, 1) \) and approaches \( y = 0 \) as \( x \to -\infty \).
Logarithmic Graphs
Logarithmic functions have the form \( f(x) = \log_a(x) \), where \( a \) is a positive real number, and \( a \neq 1 \).
Key Features:
- Vertical Asymptote: The graph has a vertical asymptote at \( x = 0 \).
- x-intercept: The x-intercept is at \( (1, 0) \).
- No y-intercept: The function is undefined for \( x \le 0 \).
For example, the function \( f(x) = \log_2(x) \) is a logarithmic function. The graph passes through \( (1, 0) \) and increases as \( x \) increases.
Relationship Between Exponential and Logarithmic Functions
Exponential and logarithmic functions are inverses of each other. The exponential function \( f(x) = a^x \) and the logarithmic function \( g(x) = \log_a(x) \) satisfy:
\( f(g(x)) = a^{\log_a(x)} = x \) and \( g(f(x)) = \log_a(a^x) = x \).
Their graphs are reflections of each other over the line \( y = x \).
Example
Consider the following functions:
- Exponential Function: \( f(x) = 2^x \)
- Logarithmic Function: \( g(x) = \log_2(x) \)
The exponential graph increases rapidly, passing through \( (0, 1) \), and approaches the horizontal asymptote \( y = 0 \). The logarithmic graph passes through \( (1, 0) \) and has a vertical asymptote at \( x = 0 \).
These functions are reflections of each other across the line \( y = x \).
Examples of Exponential and Logarithmic Graphs
Example 1: Exponential Growth
Consider the exponential function \( f(x) = 3^x \).
- Asymptote: The horizontal asymptote is at \( y = 0 \).
- y-intercept: At \( x = 0 \), \( f(0) = 3^0 = 1 \), so the y-intercept is at \( (0, 1) \).
- Growth: As \( x \) increases, \( f(x) \) grows rapidly because \( 3^x \) is an increasing function.
For example, when \( x = 2 \), \( f(2) = 3^2 = 9 \). The graph passes through points like \( (0, 1) \), \( (1, 3) \), and \( (2, 9) \), showing exponential growth.
Example 2: Exponential Decay
Consider the exponential function \( f(x) = (0.5)^x \).
- Asymptote: The horizontal asymptote is at \( y = 0 \).
- y-intercept: At \( x = 0 \), \( f(0) = (0.5)^0 = 1 \), so the y-intercept is at \( (0, 1) \).
- Decay: As \( x \) increases, \( f(x) \) decreases because \( 0 < (0.5) < 1 \).
For instance, when \( x = 2 \), \( f(2) = (0.5)^2 = 0.25 \). The graph passes through points like \( (0, 1) \), \( (1, 0.5) \), and \( (2, 0.25) \), illustrating exponential decay.
Example 3: Logarithmic Function
Consider the logarithmic function \( g(x) = \log_3(x) \).
- Asymptote: The vertical asymptote is at \( x = 0 \).
- x-intercept: At \( g(1) = \log_3(1) = 0 \), so the x-intercept is at \( (1, 0) \).
- Increasing Function: As \( x \) increases, \( g(x) \) increases because logarithmic functions grow slowly.
For example, when \( x = 3 \), \( g(3) = \log_3(3) = 1 \), and when \( x = 9 \), \( g(9) = \log_3(9) = 2 \). The graph passes through points like \( (1, 0) \), \( (3, 1) \), and \( (9, 2) \).
Example 4: Relationship Between Exponential and Logarithmic Functions
Consider the exponential function \( f(x) = 2^x \) and its inverse logarithmic function \( g(x) = \log_2(x) \).
- Reflection Property: These functions are reflections of each other across the line \( y = x \).
- Inverse Relationship: If \( f(a) = b \), then \( g(b) = a \).
For example, \( f(3) = 2^3 = 8 \) and \( g(8) = \log_2(8) = 3 \). This shows that \( f \) and \( g \) are inverse functions.
IB Mathematics AA SL Exponential & logarithmic functions Exam Style Worked Out Questions
Question
Find the range of possible values of k such that \(e^{2x}\) + In k = 3\(e^x\) has at least one real solution.
▶️Answer/Explanation
recognition of quadratic in \(e^x\)}}\){
\((e^x)^2 – 3e^x + In k(=0)\) OR \(A^2\) – 3A + In k (=0)
recognizing discriminant ≥ 0 (seen anywhere)
\((-3)^2\) – 4(1) (in k) OR 9 – In k
In k ≤ \(\frac{9}{4}\)
\(e^{9/5}\) (seen anywhere)
0 < k ≤ \(e^{9/4}\)
Question
Dominic jumps out of an airplane that is flying at constant altitude. Before opening his
parachute, he goes through a period of freefall.
Dominic’s vertical speed during the time of freefall, S , in m s−1 , is modelled by the following function.
\(S(t)=K-6-(1.2^t),t\geqslant 0\)
where t , is the number of seconds after he jumps out of the airplane, and K is a constant. A sketch
of Dominic’s vertical speed against time is shown below.
Dominic’s initial vertical speed is \(0\:ms^{-1}\)
Find the value of K . [2]
In the context of the model, state what the horizontal asymptote represents. [1]
Find Dominic’s vertical speed after 10 seconds. Give your answer in km h−1 . [3]
▶️Answer/Explanation
Ans:
(a)
0 = K – 60(1.20)
K= 60
(b)
the (vertical) speed that Dominic is approaching (as t increases)
OR
the limit of the (vertical) speed of Dominic
(c)
S=60-60\((1.2^{-10})\)
\(S=50.3096..(ms^{-1})\)
\(181(kmh^{-1})(181.144..(kmh^{-1}))\)