Common Mistakes: Functions & Modeling
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Understanding common mistakes helps you avoid them on tests and assignments. This lesson covers the most frequent errors students make with exponential and logarithmic functions, along with strategies to prevent them.
Categories of Common Mistakes
- Logarithm Property Errors: Misapplying log rules
- Domain and Range Mistakes: Forgetting restrictions
- Exponent Errors: Incorrect simplification
- Setup Errors: Wrong formula for the situation
- Calculator Mistakes: Input errors or wrong mode
Common Mistakes with Corrections
Mistake 1: Distributing Logarithms Incorrectly
WRONG: log(a + b) = log(a) + log(b)
CORRECT: log(a * b) = log(a) + log(b)
The product rule applies to multiplication inside the log, NOT addition. There is no simple rule for log(a + b).
Mistake 2: Confusing Power and Product Rules
WRONG: log(x^3) = (log x)^3
CORRECT: log(x^3) = 3 * log(x)
The exponent comes down as a coefficient, not as an exponent on the entire logarithm.
Mistake 3: Forgetting Domain Restrictions
WRONG: Solving log(x-2) = 3 and accepting x = -1
CORRECT: x - 2 must be positive, so x > 2. Check all solutions!
Logarithms are only defined for positive arguments. Always verify solutions satisfy domain restrictions.
Mistake 4: Mishandling Negative Exponents
WRONG: 2^(-3) = -8
CORRECT: 2^(-3) = 1/2^3 = 1/8
A negative exponent means reciprocal, not negative result. Exponential functions are always positive.
Mistake 5: Wrong Base with Natural Log
WRONG: ln(x) = log_10(x)
CORRECT: ln(x) = log_e(x) where e = 2.718...
Natural log uses base e. On calculators, "log" typically means base 10, while "ln" means base e.
Mistake 6: Incorrect Rate in Continuous Compounding
WRONG: A = Pe^(5t) for 5% interest
CORRECT: A = Pe^(0.05t) for 5% interest
Always convert percentages to decimals before using in formulas.
Mistake 7: Canceling Logs Incorrectly
WRONG: log(x)/log(y) = x/y
CORRECT: log(x)/log(y) = log_y(x) (change of base)
You cannot cancel the "log" part. Use the change of base formula instead.
Mistake 8: Confusing Growth Rate and Factor
WRONG: "Grows by 5% per year" means A = A_0 * (5)^t
CORRECT: "Grows by 5% per year" means A = A_0 * (1.05)^t
The growth factor is 1 + rate. For decay, the factor is 1 - rate.
Practice: Spot the Error
Identify the mistake in each problem and provide the correct solution.
Problem 1: A student solved log_2(8) + log_2(4) = log_2(12). Is this correct?
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Error: Used addition instead of multiplication inside log.
Correct: log_2(8) + log_2(4) = log_2(8 * 4) = log_2(32) = 5
Problem 2: To solve 5^x = 20, a student wrote x = 20/5 = 4. Find the error.
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Error: Treated exponent as multiplication.
Correct: Take log of both sides: x = log(20)/log(5) = 1.861
Problem 3: For the equation ln(x) + ln(x-4) = ln(5), a student found x = 5 and x = -1. Which, if any, are valid?
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Answer: Only x = 5 is valid. x = -1 makes both ln(x) and ln(x-4) undefined.
Check: x must be > 4 for both logarithms to be defined.
Problem 4: A student simplified e^(ln 5 + ln 3) as e^(ln 5) + e^(ln 3) = 5 + 3 = 8. Correct?
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Error: Cannot distribute e^ over addition in exponent.
Correct: e^(ln 5 + ln 3) = e^(ln 15) = 15
Problem 5: A $10,000 investment at 6% compounded annually for 5 years: A = 10000(1.6)^5. Find the error.
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Error: Used 1.6 instead of 1.06 as the growth factor.
Correct: A = 10000(1.06)^5 = $13,382.26
Problem 6: A student wrote log_3(27) - log_3(9) = log_3(27-9) = log_3(18). Find the error.
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Error: Subtracted inside the log instead of dividing.
Correct: log_3(27) - log_3(9) = log_3(27/9) = log_3(3) = 1
Problem 7: Solving 3^(2x) = 9^(x+1): A student wrote 2x = x + 1, so x = 1. Check this work.
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Error: Did not convert to same base first.
Correct: 3^(2x) = (3^2)^(x+1) = 3^(2x+2), so 2x = 2x + 2, which has no solution.
Problem 8: If a population doubles every 5 years, a student wrote P(t) = P_0 * 2^(5t). Find the error.
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Error: The exponent should be t/5, not 5t.
Correct: P(t) = P_0 * 2^(t/5)
Problem 9: A student simplified (log x)^2 = 2 log x. Is this valid?
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Error: (log x)^2 is not the same as log(x^2).
Correct: (log x)^2 cannot be simplified this way. log(x^2) = 2 log x, but (log x)^2 stays as is.
Problem 10: To find when $5000 becomes $8000 at 4% continuous interest, a student wrote: t = ln(8000/5000)/4. Find the error.
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Error: Used 4 instead of 0.04 for the rate.
Correct: t = ln(1.6)/0.04 = 11.78 years
Problem 11: A student solved log(x) = -2 as x = -100. Correct?
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Error: Confused the negative with the argument.
Correct: log(x) = -2 means x = 10^(-2) = 0.01
Problem 12: For half-life of 10 years: A student wrote A = A_0 * (1/2)^(10t). Find the error.
Show Answer
Error: The exponent should be t/10, not 10t.
Correct: A = A_0 * (1/2)^(t/10)
Check Your Understanding
Answer these questions to solidify your understanding of common errors.
- Why can you not simplify log(a + b) into log(a) + log(b)?
- What should you always check after solving a logarithmic equation?
- How do you correctly interpret "decreases by 15% per year" in an exponential model?
- What is the difference between log(x^2) and (log x)^2?
Next Steps
- Create a personal "error log" of mistakes you tend to make
- Practice problems slowly, checking each step
- Review logarithm properties until they become automatic
- Take the Unit Quiz to test your understanding