Podcast
Questions and Answers
When solving logarithmic equations, what is the primary reason for checking for extraneous roots?
When solving logarithmic equations, what is the primary reason for checking for extraneous roots?
- To avoid taking the logarithm of a negative number or zero. (correct)
- To find all possible solutions to the equation.
- To ensure the solution satisfies the original equation.
- To simplify the algebraic expression.
The notation 'M1' in a marking scheme typically represents marks awarded for the correct answer, even without showing the method.
The notation 'M1' in a marking scheme typically represents marks awarded for the correct answer, even without showing the method.
False (B)
In question Q1, what property of logarithms is used to combine $\log(x-1) + \log(x+2)$ into a single logarithmic term?
In question Q1, what property of logarithms is used to combine $\log(x-1) + \log(x+2)$ into a single logarithmic term?
log(a) + log(b) = log(ab)
According to the decay equation given, $A_0$ represents the ______ amount.
According to the decay equation given, $A_0$ represents the ______ amount.
Match the marking scheme notations with their meaning:
Match the marking scheme notations with their meaning:
In the half-life decay equation $A(t) = A_0(\frac{1}{2})^{t/h}$, what does 'h' represent?
In the half-life decay equation $A(t) = A_0(\frac{1}{2})^{t/h}$, what does 'h' represent?
The solution $x = -50$ is an extraneous root for the equation $\log\sqrt[3]{x^2 + 48x} = \frac{2}{3}$.
The solution $x = -50$ is an extraneous root for the equation $\log\sqrt[3]{x^2 + 48x} = \frac{2}{3}$.
What two methods are specified in the content for solving quadratic equations in the context of logarithmic equations?
What two methods are specified in the content for solving quadratic equations in the context of logarithmic equations?
The half-life of the radioactive substance, calculated in the provided example, is approximately:
The half-life of the radioactive substance, calculated in the provided example, is approximately:
According to the calculations, a radioactive substance initially at 50 mg will decay to 0.5 mg in approximately 232.5 minutes.
According to the calculations, a radioactive substance initially at 50 mg will decay to 0.5 mg in approximately 232.5 minutes.
In the equation $4^x = 6$, what is the approximate value of x?
In the equation $4^x = 6$, what is the approximate value of x?
To solve for the half-life, logarithms are applied to ______ sides of the equation.
To solve for the half-life, logarithms are applied to ______ sides of the equation.
In the exponential decay equation $A(t) = A_0 (1/2)^{(t/h)}$, what does 'h' represent?
In the exponential decay equation $A(t) = A_0 (1/2)^{(t/h)}$, what does 'h' represent?
When solving the equation $4^x = -3$, the solution exists within the real number system.
When solving the equation $4^x = -3$, the solution exists within the real number system.
Rewrite the equation $4^x - 3 - 18(4^{-x}) = 0$ by taking $4^x$ common.
Rewrite the equation $4^x - 3 - 18(4^{-x}) = 0$ by taking $4^x$ common.
Match the variable with its value:
Match the variable with its value:
Flashcards
Marking Scheme
Marking Scheme
System to allocate marks for methods, application, and reasoning.
Extraneous Root
Extraneous Root
An invalid solution that does not satisfy original equations.
Logarithmic Identity
Logarithmic Identity
Logarithm of a product can be expressed as a sum: log(a) + log(b) = log(ab).
Decay Equation
Decay Equation
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Half-Life
Half-Life
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Quadratic Factorization
Quadratic Factorization
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Valid Solutions
Valid Solutions
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Logarithm Undefined
Logarithm Undefined
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Logarithm
Logarithm
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Calculate decay time
Calculate decay time
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Change of variable
Change of variable
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Quadratic equation
Quadratic equation
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Negative solutions
Negative solutions
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Exponential growth/decay
Exponential growth/decay
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Study Notes
Test Marking Scheme - Chapter 7
- General marking notations
- M: Method marks
- A: Application marks
- R: Reasoning marks
- N: Correct answer with no working shown
Question 1 (k=30) Part a
- Isolate logarithmic terms on one side
- Use the rule log(a) + log(b) = log(ab)
- Solve the quadratic equation (x² + x - 12 = 0) to find x values
- x = 3 or x = -4
- Determine extraneous roots
- x = -4 is an extraneous root as log(x-1) and log(x+2) are undefined for a negative input
Question 1 (k=30) Part b
- Use the rule log(ab) = b log a
- Solve the quadratic equation (x² + 48x - 100 = 0) to find x values
- x = 2 or x = -50
Question 2 (A=30) Part a
- Decay equation: A(t) = Aâ‚€(1/2)^(t/h)
- Given Aâ‚€ = 50 mg
- Given half-life (h)
- Given A(t) = 41 mg at t = 10 min
- Substitute values into the equation to solve for h
- 41 = 50 * (1/2)^(10/h)
- Take log on both sides
- Solve for h
Question 2 (A=30) Part b
- Given 1% of 50mg = 0.5 mg
- Given A(t) = 0.5 mg
- Calculate the time (t) it takes to decay to 0.5 mg
- Apply the decay equation: A(t) = 50(1/2)^(t/hâ‚€)
- Solve for the variable t
- Apply logarithms to solve for t
Question 3 (T=20) Part e
- Given equation 3^(x+1) + 56(3^(-x)) = 0
- Rewritten to: t² + t + 56 = 0 (substitute 3^x = t)
- Determine the existence of real roots to the quadratic
- Quadratic equation has no real roots, thus no valid solution
Question 3 (T=20) Part f
- Given equation 4^x = 3+18(4^-x)
- Rewrite the equation (4^(2x) - 3(4^x) - 18 = 0)
- Substitute 4^x = t
- Solve t² - 3t -18 = 0 (quadratic)
- Find the values of t using factoring and thus x
Question 4 (T=20) Part a
- Basic logarithm rule: log(a²bc)=2loga + logb + logc
Question 4 (T=20) Part b
- Basic logarithm rules: log(A/√m) = log A - log(m^(1/2) ) = logA - (1/2) log m
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