Why students lose marks in exponents and logarithms and it is almost always a forgotten domain check
In 25 years of teaching, the exponents and logarithms mistake I see most often has nothing to do with the laws themselves. A student solves log3x + log3(x – 6) = 3 correctly, combines the two logs, forms a quadratic, factorises it, and finds two roots. Both get written down as the final answer. One of those roots makes log3(x – 6) undefined, because it requires taking the logarithm of a negative number, and the mark scheme rejects that solution outright. The algebra was perfect. The domain check that should have happened before the answer was written down never did. The same gap appears with exponential equations written as quadratics in disguise, where students who do not substitute u = e^x or u = 3^x lose the structure of the equation entirely and cannot recover both solutions. These worksheets train the habit of stating the domain or the substitution before starting the algebra, and checking every solution against it afterwards, every single time.
Recognition Training
Exponents and Logarithms - Mistake Analysis - Set I
Six questions opening with solving 2 to the power x plus 1 equals 32 by equating exponents, then a logarithmic equation log3x + log3(x - 4) = 3 requiring a domain check and rejection of an extraneous root, expanding a logarithmic expression with a square root and a cube using all three laws together, solving e^2x - 5e^x + 6 = 0 by substitution, a compound interest problem finding the minimum number of years, and a hence question linking log4 8 = 3/2 to solving a logarithmic equation.
Exponents and Logarithms - Mistake Analysis - Set II
Six questions on solving an exponential equation by quadratic substitution, analysing the vertical asymptote, range and x-intercept of a logarithmic function, proving the change of base identity loga x equals 1 over logx a, solving ln(x + 2) + ln(x - 1) = ln(4x - 2) with a domain check and rejection of an extraneous root, an exponential growth model for a population, and a pair of simultaneous equations in two different bases solved by substitution.
Exponents and Logarithms - Mistake Analysis - Set III
Six questions opening with a non calculator evaluation of a logarithmic expression using the laws of logarithms, a logarithmic equation log2(x + 3) = 3 - log2x that rearranges into a quadratic, expressing log10 of the square root of 24 in terms of given constants p and q, a two variable compound interest problem finding both the principal and the rate by dividing two equations, solving 2log5x - log5(x - 4) = 2 with verification of both solutions, and a telescoping product of logarithms across six different bases.
The 4 Patterns Behind Every Lost Mark
Logarithms Do Not Distribute Over Addition, Subtraction or Multiplication
Subtraction of two logarithms corresponds to division of their arguments, not subtraction of the arguments. Addition of two logarithms corresponds to multiplication of their arguments, not addition. Students who write log(A - B) as log A minus log B, or who treat log A times log B as log(AB), are applying rules that do not exist and will reach an expression that cannot be simplified.
Forgetting to Check the Domain and Reject Extraneous Solutions
Every logarithmic term in an equation has its own domain, and that domain does not change once the logarithms are combined. A quadratic formed from a logarithmic equation will often produce one root that satisfies the combined equation but makes a logarithm in the original equation undefined. Every solution must be checked against the domain of the original equation, not the combined one, and any root that fails must be explicitly rejected.
Missing the Required Substitution in Exponential Equations That Are Quadratics in Disguise
When a^2x and a^x both appear in the same equation, the equation is a quadratic in u = a^x. Students who treat a^2x as 2a^x, confusing the exponent rule with the rule for differentiating e^2x, collapse the equation into something that can be solved but produces only one of the two correct solutions. The substitution u = a^x must come first.
Misapplying the Power Rule When the Exponent is a Fraction or Negative
The power rule states that log of A to the power n equals n times log A, and this applies just as much when n is a fraction or negative as when it is a positive integer. Students who write log of the square root of y as 2 log y instead of one half log y, or who mishandle a negative exponent when isolating a base, produce an answer that is wrong by a multiplicative factor even though every other step was correct.
The Full Diagnostic Path
- 75 original exam-style questions across exponents, logarithms and their applications
- Full worked solutions with M1/A1/R1 IB mark scheme annotations
- Mistake analysis on every question targeting domain errors, distribution errors, and missing substitutions
- Sections: laws of exponents, laws of logarithms, exponential and logarithmic equations, change of base, compound interest and growth models
- IB Examiner commentary per section on where marks are most commonly lost
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