4.4 Laws of Logarithms
Ideas in this lesson
- Prove log laws and diagnose false identities
- Evaluate combined and iterated logarithms exactly
- Expand logarithms without losing the real domain
- Condense logarithms and retain excluded inputs
- Change bases, graph logarithms and prove related identities
- Recover log-linear models and interpret logarithmic differences
- Prove equivalent horizontal and vertical transformations
Prove log laws and diagnose false identities · Worked example
Distinguish \(\dfrac{\log A}{\log B}\) from \(\log\dfrac{A}{B}\). Evaluate both at \(A=10^3,\ B=10^2\) and state when the first expression is defined. Written work is required; selecting an answer alone is not a complete response. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: \(\dfrac{3}{2}\) versus \(1\); the quotient of logs additionally excludes \(B=1\).
The numerator and denominator of the quotient of logs are \(3\) and \(2\).
The log of the quotient is \(\log(\dfrac{10^{3}}{10^{2}})=\log10=1\).
The first requires \(A>0,\ B>0,\ B\ne1\); the second only requires its ratio positive, or positive separate arguments if using the quotient law.
These are different operations, not alternate notations.
Written-work review:
The log of the quotient is \(\log(\dfrac{10^{3}}{10^{2}})=\log10=1\).
The first requires \(A>0,\ B>0,\ B\ne1\); the second only requires its ratio positive, or positive separate arguments if using the quotient law.
These are different operations, not alternate notations.
Written-work review:
- Accept mathematically equivalent derivations and valid alternative counterexamples. Every requested conclusion, original-domain restriction, justification, precision and unit requirement still applies; the listed solution is one acceptable route.
- Required written evidence: The numerator and denominator of the quotient of logs are \(3\) and \(2\).
- Required written evidence: The log of the quotient is \(\log(\dfrac{10^{3}}{10^{2}})=\log10=1\).
- Required written evidence: The first requires \(A>0,\ B>0,\ B\ne1\); the second only requires its ratio positive, or positive separate arguments if using the quotient law.
- Required written evidence: These are different operations, not alternate notations.