VCE · Maths Methods · Functions, Graphs & Transformations live from the app

Logarithmic Functions and Log Laws

A logarithm is just an exponent asking to be found — log base a of b answers 'what power turns a into b?'. Because logs invert exponentials, their graph is the exponential graph reflected in the line y = x. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.

maths methods · functions, graphs & transformations · logarithmic functions and log lawsdrag it · it is yours
in the wild

Functions work like a takeaway order.

Every function here is a known shape moved around: shift, stretch, flip, then read off where it went. Polynomials, exponentials, sine curves — a handful of parent graphs under transformation; inverses run the moves backwards, composites stack them.

Orders queue at the counter — each one a different input waiting for the kitchen.

The kitchen applies its rule, turning each order into exactly one meal on the pass.

Every order leaves as one dish — same order always gives the same output.

what examiners catch — Students drop marks by misapplying log laws under addition (log(a+b) does not equal log a plus log b) and by forgetting the domain restriction that you can only take logs of positive numbers.
what you leave with

five things, not forty.

what's underneath

nothing here is a standalone fact.

knowhere maps every concept to what it rests on and what rests on it — 2 underneath this one, 2 built on top. Each one says why, in a sentence, not as an arrow on a diagram.

this conceptlogarithmic functions and log lawsLogarithm base a of x asks what power you raise a to get x, so log laws mirror index laws, and the graph of log base a reflects the graph of a^x across the diagonal line y equals x.
sits under itexponential functions and ebecause a log is an exponential read backwards
sits under itinverse functionsbecause the log IS the inverse of the exponential — the general machinery first
built on itmodelling with functionsbecause decibels, pH and seismic scales are log models — the laws come before the modelling
built on itsolving exponential and logarithmic equationsbecause solving requires the inverse move
the rest of functions, graphs & transformations

7 more, same treatment.

Each one is its own knowscape in the app — built for how a particular student takes things in, not one explanation handed to everybody.

composite functionsin the appinverse functionsin the appcircular functions and their graphslive →modelling with functionsin the apppolynomial functions and their graphsin the apptransformations of graphsin the appexponential functions and ein the app
this is one of 865

they don't read it. they run it.

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knowherevce maths methodslogarithmic functions and log laws