4.4 Applications of Indices, Surds and Logarithms
Mathematical models involving indices and logarithms are widely used to represent real-life phenomena such as biological population growth, pest infestations, radioactive decay, financial investments, and electrical currents. By applying algebraic rules of indices, equating powers of the same base, or taking logarithms on both sides of an equation, unknown variables such as time or final quantities can be solved accurately.
Learning Objectives
- Apply the laws of indices and logarithms to solve real-life application problems involving exponential and logarithmic equations.
- Model and solve problems across various disciplines such as biology, finance, and physics following the four-step problem-solving process.
- Evaluate initial values by substituting \(t = 0\) or \(n = 0\), and find unknown elapsed time by solving index or natural logarithmic equations.
The 4-Step Problem-Solving Cycle in Exponential & Logarithmic Applications
A clear step-by-step schematic for setting up, solving, and concluding real-world application problems using indices and logarithms.
Four Connected Stages
Understanding the problem: Identify the given formula, variables such as \(t\), \(n\), \(A\), \(I\), \(W\), and the target unknown.
Planning the strategy: Choose direct substitution if time is given, or index equating / logarithms if time is unknown.
Implementing the strategy: Execute calculations step-by-step using \(a^x = a^y \implies x = y\) or \(x = \dfrac{\log b}{\log a}\).
Making a conclusion: State the answer with proper context and units, and check validity by substitution.
Short Explanation
Applying indices and logarithms to real-life situations allows us to determine future quantities or calculate the exact time needed for an event to take place.
1. Finding an Initial Value
Set the time parameter \(t\) or \(n\) to \(0\). Since any non-zero base raised to the power of \(0\) is \(1\), that is \(a^0 = 1\), the initial quantity is readily obtained.
2. Finding the Value After a Specified Time
Directly substitute the value of \(t\) or \(n\) into the exponent and evaluate the resulting numerical expression.
3. Finding the Elapsed Time
Substitute the target amount into the dependent variable, then isolate the exponential term. If both sides can be rewritten with the same base, equate the indices directly: \(a^u = a^v \implies u = v\).
4. Using Logarithms for Unknown Exponents
If the bases cannot be rewritten with a common base, or if base \(e\) is involved, take common logarithms \(\log\) or natural logarithms \(\ln\) on both sides and apply the power law \(\log a^b = b \log a\).
| Model | Formula | Key Application |
|---|---|---|
| Grasshopper infestation | \(A(n) = 1\,000 \times 2^{0.2n}\) | Area in acres after \(n\) weeks |
| Electrical current decay | \(I = 32 \times 4^{-t}\) | Current in amp after \(t\) seconds |
| Bank savings growth | \(W = 1\,000(1.09)^t\) | Accumulated money in RM after \(t\) years |
| Uranium radioactive decay | \(W(t) = 50 \times 2^{-0.0002t}\) | Mass in grams after \(t\) years |
| Bacterial growth | \(J = 25 \times e^{0.1t}\) | Mass in grams after \(t\) hours |
Grasshopper infestation area
\(A(n) = 1\,000 \times 2^{0.2n}\)\(A\) is in acres and \(n\) is the number of weeks.
Bacterial mass growth
\(J = 25 \times e^{0.1t}\)\(J\) is in grams and \(t\) is the time in hours.
Try to Answer First
Answer in your mind, then press "Check Answer".
An electric current flowing through a circuit \(t\) seconds after being switched off is given by \(I = 32 \times 4^{-t}\text{ amp}\). Determine the current when the switch is first turned off.
The mass of a bacterial culture \(t\) hours after observation begins is given by \(J = 25 \times e^{0.1t}\text{ grams}\). Show that the time required for the mass to double to \(50\text{ grams}\) is \(10 \ln 2\text{ hours}\).
A grasshopper infestation spreads across an area given by \(A(n) = 1\,000 \times 2^{0.2n}\text{ acres}\), where \(n\) is the number of weeks. Calculate how many weeks are needed for the infestation to cover an area of \(8\,000\text{ acres}\).
Common Mistakes
- Multiplying the coefficient with the exponential base before applying the power. For example, in \(A = 1\,000 \times 2^{0.2n}\), students mistakenly multiply \(1\,000 \times 2\) to get \(2\,000^{0.2n}\). The order of operations requires exponentiation before multiplication.
- Forgetting that \(a^0 = 1\) when determining initial values. Some students write \(A(0) = 0\) instead of \(1\,000 \times 2^0 = 1\,000 \times 1 = 1\,000\).
- Incorrect division when isolating the exponential term. When solving \(0.5 = 32 \times 4^{-t}\), students sometimes subtract \(32\) instead of dividing by \(32\), or mistakenly write \(\dfrac{32}{0.5} = 64\) instead of \(\dfrac{0.5}{32} = \dfrac{1}{64}\).
- Mismanaging negative powers. Confusing \(4^{-t} = 4^{-2}\) with \(t = -2\) instead of \(-t = -2 \implies t = 2\).
- Omitting units in the final conclusion. Writing a bare numerical value like \(15\) or \(3\) instead of stating the unit clearly, such as \(15\text{ weeks}\) or \(3\text{ seconds}\).
Concept Misunderstandings
\(\log(a \times b) = \log a \times \log b\).
The correct law is \(\log(a \times b) = \log a + \log b\).
Take logarithms of \(1\,000 \times 2^{0.2n} = 8\,000\) and write \(\log(1\,000) \times 0.2n \log 2\).
Isolate \(2^{0.2n} = 8\) first, or use the product expansion \(\log 1\,000 + 0.2n \log 2\). Isolating the base first is much simpler and prevents algebraic errors.
Equations with base \(e\) must use common \(\log_{10}\).
For equations containing base \(e\), such as \(e^{0.1t} = 2\), taking the natural logarithm \(\ln\) directly simplifies the expression since \(\ln(e) = 1\), yielding \(0.1t = \ln 2\).
Summary
- Real-life processes of growth and decay are modeled using exponential functions of the general form \(y = k \cdot a^{ct}\) or \(y = k \cdot e^{ct}\).
- Initial quantities correspond to \(t = 0\) or \(n = 0\), where \(a^0 = 1\).
- To find an unknown duration or time exponent, isolate the base term containing the unknown exponent.
- If both sides share an obvious common base, equate indices: \(a^u = a^v \implies u = v\).
- If bases differ or involve base \(e\), apply common logarithms \(\log\) or natural logarithms \(\ln\) and apply the power rule \(\log a^b = b \log a\).
- Always structure solutions using the 4-step problem-solving framework and include appropriate units in the final conclusion.
Short Activity
Objective: Apply indices and logarithms to solve real-world application problems.
A. Objective Quiz
1 An electric current flows according to the formula \(I = 32 \times 4^{-t}\text{ amp}\), where \(t\) is the time in seconds after the switch is turned off. What is the current flowing in the circuit after \(1\text{ second}\)?
2 The mass of a bacterial culture increases according to \(J = 25 \times e^{0.1t}\text{ grams}\). Which of the following expressions represents the exact time \(t\), in hours, required for the mass to reach \(50\text{ grams}\)?
B. Fill in the Blanks
3 For a grasshopper infestation area modeled by \(A(n) = 1\,000 \times 2^{0.2n}\text{ acres}\), the area of infestation at the beginning is ________ acres.
4 The time taken for the electric current \(I = 32 \times 4^{-t}\text{ amp}\) to reach \(0.5\text{ amps}\) is ________ seconds.
C. Matching / Drag and Drop
Match each step of the problem-solving framework and application concept to its correct mathematical implementation or description. On a phone, tap a term first, then tap the matching box.
