The average rainfall in Phoenix is 8.29 inches per year. The table shows recent data on the difference in annual rainfall from the average. Phoenix Annual Total RainfallYearRainfall compared to average yearly rainfall2008+6.57 inches2009–2.68 inches2010+12.26 inches2011–4.38 inches2012–4.46 inchesWhich list represents the years from driest to wettest?2010, 2008, 2009, 2011, 20122011, 2012, 2009, 2008, 20102012, 2011, 2009, 2008, 20102009, 2011, 2012, 2008, 2010

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1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: