12E - Logarithmic equations
Probability and logarithms: Have you been paying attention?
On the show Have you been paying attention? the host, Tom Gleisner, has a segment called "over or under" where the contestants are shown a short video from RBT (random breath testing). Have a go at playing "over or under" now, record your answers as "over" or "under" before the reveal!
■ Warning! The video contains coarse language. |
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If the clips are selected selected clips that each contestant has a 50% chance of guessing correctly if the person is over or under the blood alcohol limit (0.05). In total there are 5 contestants on the show.
(a)
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What is the probability that all 5 contestants guess correctly?
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(b)
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What is the probability that exactly 2 of the contestants guess correctly?
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(c)
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What is the probability that two or more contestants guess correctly?
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(d)
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What is the probability that exactly 2 contestants guess correctly, given that 2 or more guessed correctly?
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(e)
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Imagine you are the contestant. What is the least number of clips that must be played to ensure the probability of getting at least one correct is over 95%?
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We can use the laws and observations regarding logarithms in Section 12D to simplify and solve logarithmic equations. The following examples demonstrate the process of solving logarithmic equations.
12E - Example 1: Solving logarithmic equations
Solve each of the following equations for \(x\).
(a) \(log_3(x+1)=2\) (b) \(log_5(x^2)=4\) |
12E - Example 1: Video solution
12E - Example 1: Practice
Question 1: ABC Question 2: ABC 12E - Example 1: Solutions
Question 1: ABC Question 2: ABC |
12E - Example 2: Solving logarithmic equations
Solve the following equation for \(x\)
\(log_2(x+4)+log_2(x-4)=5\)
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12E - Example 2: Video solution
12E - Example 2: Practice
Question 1: ABC Question 2: ABC 12E - Example 2: Solutions
Question 1: ABC Question 2: ABC |
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Additional Exercises
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Topic Worksheets
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Other Resources
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Question 1 [SQA, 2017]
Solve the following equation for \(a\): \[log_a(36)-log_a(4)=\frac{1}{2}\] |
Solutions:
Question 1: \[log_a(9)=\frac{1}{2}\] \[\therefore a^{\frac{1}{2}}=9 \therefore a=81\] |