Logarithm mistakes often look like calculator mistakes but begin earlier: an invalid law is used, a domain restriction is ignored, or the meaning of a transformed quantity is lost. Diagnosing the first unsafe line is more useful than merely replacing the final answer.
1. Error: splitting a logarithm of a sum
Incorrect: log(a+b)=loga+logb.
Repair: product and quotient laws apply to multiplication and division, not addition. Keep log(a+b) intact unless the inside expression can first be factorised or otherwise transformed validly.
2. Error: forgetting the argument must be positive
For ln(x−3), require x>3. A later algebraic root x=1 cannot be accepted because the original expression is not real-defined there.
3. Error: confusing log base 10 and natural log
If ln a=2, then a=e². If log₁₀a=2, then a=100. The inverse operation must use the same base as the logarithm.
4. Error: treating an intercept as the original constant
In a linearised power model log y=n log x+log a, the vertical intercept is log a, not a. Recover a by applying the inverse logarithm.
5. Error: moving an exponent incorrectly
log(x³)=3logx for x>0 in the standard real-log setting used here. The exponent becomes a multiplier; it does not become log3.
6. Error: taking logs before isolating a useful exponential form
For 4(3^x)+2=38, first isolate 3^x: 4(3^x)=36, so 3^x=9 and x=2. Taking logs of the original sum creates no valid shortcut because log(A+B) does not split.
7. Error: substitution produces an impossible value
If u=2^x, then u>0. A transformed quadratic may produce u=−3, but that root cannot correspond to any real x because 2^x is positive.
8. Error: rounding logs too early
For x=ln7/ln3, retaining calculator precision until the final step gives a more reliable approximation than rounding each logarithm first.
9. Error: reading transformed graph axes as original variables
A point (1,2) on a graph of ln y against ln x means ln x=1 and ln y=2. It does not mean x=1 and y=2. The original values are x=e and y=e².
10. Error: believing a straight transformed graph proves a universal law
For exact school-model data, a straight transformation can be part of the stated relationship. For measured data, approximate linearity supports a model over the observed range but does not by itself prove causation or indefinite validity.
11. A diagnostic sequence
- Check every logarithm argument.
- Check whether each log law matches multiplication, division or a power.
- Check the logarithm base.
- Check transformed variable meanings.
- Check candidates in the original equation.
12. Practice diagnosis
- A student writes ln(x+2)=lnx+ln2. Identify the error.
- A student solves ln(x−1)=0 and reports x=0. Repair the solution.
- If log₁₀a=−2, find a.
- In log₁₀y=3log₁₀x+0.4, identify the power and the multiplicative constant.
- If u=5^x and algebra gives u=1 or −4, which value can continue?
13. Answers
- There is no logarithm sum law; keep ln(x+2) intact.
- ln(x−1)=0 means x−1=e⁰=1, so x=2.
- 10^−2=0.01.
- Power 3; constant 10^0.4.
- Only u=1 because 5^x>0 for real x.
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