MHF 4U Advanced Functions
Exponential and Ligarithmic Functions Test 14
Category
Total Marks
Knowledge and Understanding
19
Application
19
Thinking and Inquiry
13
Communication
12
Marks Obtained
Part A: Multiple Choice
1. Convert the following exponential equation to logarithmic form:
3x = 243 [2K]
(a) logx 243 = 5
(c) log3 x = 243
(b) log3 243 = x
(d) logx 3 = 243
2. What type of asymptote is present in a logarithmic function? [2K]
(a) No asymptote
(b) Vertical asymptote
(c) Horizontal asymptote
(d) Oblique asymptote
3. If loga x = 16, determine the value of loga √𝑥. [2T]
(a) 4
(b) 8
(c) 16 (d) 32
4. What is the domain of the logarithmic function y = log3(3x + 2) – 1?
[2A]
(a) {x ∈ ℝ | x > 0}
(b) {x ∈ ℝ | x > 2}
2
(c) {x ∈ ℝ | x > – 2}
(d) {x ∈ ℝ | x > – }
3
Part B: Full Solutions
5. Evaluate the following expressions: [8K]
(a) log82
(b) log 1 3 − log 1 12
2
(c) 10log 7+ log 5
(d) logb 1, b > 0
2
6. Express as a single logarithm and simplify: [4K]
3log2(a + b) + 2log4(a + b) – log2c, a + b > 0, c > 0
7. Consider the equation pH = – log [H+], where [H+] is the
concentration of hydrogen ions in mol/L. What is the difference in pH
between a solution with [H+] = 0.006 mol/L and a solution with [H+] =
0.0001 mol/L? [1C 3K]
8. Solve the following logarithmic and exponential equations for x:
[17A]
𝑥
(a) 76 = √7
(b) log3 4x + log3 5 – log3 2 = 4
2
1 2𝑥−𝑥
(c) ( )
2
=8
(d) log3 (x – 2) + log3 x = 1
(e) 64𝑥 − 2 = 8𝑥 ∙ √8
9. Jordan is a cryptographer who Hs been trying to solve the following
code: NWP BDLTB. He has given the key for the code, but he has
trouble working with logarithms. Given the key below, what is the code?
[3T 2C]
Key: Take the value of x in the equation log7 (log2 (log3 x))) = 0 and
shift each letter in the code that many places back in the alphabet.
10. Consider the logarithmic function f(x) = – 2log (2x – 8). [8T]
(a) State the x-intercepts, y-intercept, and asymptote.
(b) Sketch the graph of y = f(x).
(c) Determine the values of x for which f(x)≥ 0.
11. What is the relationship between an exponential function and a
logarithmic function? What is the relationship between their domain and
range? State two other characteristics between the two functions. [4C]
12. State the transformations needed to transform the graph of y = 1 –
4log(x + 3) – 1 into the graph of y = – 2log(x + 5) + 2. [3C]
Bonus Question
13. Explain why logx xm – 1 + 1= m, x > 0