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Concept · Functions & Sequences
Power, exponential, logarithmic functions
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A Functions & Sequences set, untimed
Named traps
7 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- Different log bases not convertedCompares logs of different bases without change-of-base conversion.→
- Exp-to-log variable swapMishandles converting an equation like 3a=5b to logs, attaching the log to the wrong variable.→
- Exponential term mis-estimatedEstimates an exponential term's magnitude incorrectly when comparing a, b, c.→
- Origin-passing exponent conditionIgnores the sign/zero condition on the exponent that controls passing through the origin.→
- Power vs exponential formTreats a power function as exponential (or vice versa) when fitting through a point.→
- Power-function coefficient not 1Forgets a genuine power function needs coefficient 1, accepting an extraneous parameter value.→
- ln near 1 sign misjudgedMisjudges the sign/magnitude of ln of a value close to 1 when ordering expressions.→
Key formulas
9- Functiony = f(x), domain D, range Requal functions require same domain AND same rule
- Domain conditionsdenominator ≠ 0even-root radicand ≥ 0log argument > 0base of non-integer power > 0intersect all conditions
- Even: f(−x) = f(x) (symmetric about y-axis)Odd: f(−x) = −f(x) (symmetric about origin)parity requires a domain symmetric about 0
- Power functiony = xa (coefficient must be 1)
- Exponentialy = ax (a>0, a≠1)natural base e
- Logarithmy = loga x, inverse of axln = loge, lg = log10
- Log rulesloga(MN)=loga M+loga Nloga(M/N)=loga M−loga Nloga(Mk)=k·loga M
- Change of baseloga b = logc b / logc aidentities aloga N=N, loga(an)=n
- Periodicityf(x + T) = f(x)
Shaky underneath?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
FoundationsPrecalculus bridgeAlgebra 2 foundations