Exponential Logarithmic Functions Dr Carol A Marinas Table of Contents Exponential Functions Logarithmic Functions Converting between Exponents and Logarit hms Properties of Logarithms Exponential and Logarithmic Equations General Form of Exponential Function Domain All reals Range y 0 x intercept None y intercept 0 1 y bx where b 1 General Form of Exponential Function y b x c d where b 1 c moves graph left or right opposite way d move graph up or down expected way So y 3 x 2 3 moves the graph 2 units to the left and 3 units up 0 1 to 2 4 Relationships of Exponential y bx Logarithmic y logbx Functions y bx Domain All Reals Range y 0 x intercept None y intercept 0 1 y logbx is the inverse of y bx Domain x 0 Range All Reals x intercept 1 0 y intercept None Relationships of Exponential y bx Logarithmic y logbx Functions Converting between Exponents Logarithms BASEEXPONENT POWER 42 16 4 is the base 2 is the exponent 16 is the power As a logarithm logBASEPOWER EXPONENT log 4 16 2 Logarithmic Abbreviations log10 x log x Common log loge x ln x Natural log e 2 71828 Properties of Logarithms logb MN logbM logbN Ex log4 15 log45 log43 logb M N logbM logbN Ex log3 50 2 log350 log32 logbMr r logbM Ex log7 103 3 log7 10 logb 1 M logbM 1 1 logbM logbM log11 1 8 log11 8 1 1 log11 8 log11 8 Properties of Logarithms Shortcuts logb1 0 because b0 1 logbb 1 because b1 b logbbr r because br br blog bM M because logbM logbM Examples of Logarithms Simplify log 7 log 4 log 2 log 7 4 log 14 2 Simplify ln e2 2 ln e 2 logee 2 1 2 Simplify e 4 ln 3 3 ln 4 e ln 34 ln 43 e ln 81 64 e loge 81 64 81 64 Change of Base Formula log712 log 7 logam log 12 logbm OR logab log712 ln 12 ln 7 Exponential Logarithmic Equations If logb m logb n then m n If log6 2x log6 x 3 then 2x x 3 and x 3 If bm bn then m n If 51 x 5 2x then 1 x 2x and x 1 If your variable is in the exponent Isolate the base exponent term Write as a log Solve for the variable Example 4x 3 7 log 4 7 x 3 and 3 log 4 7 x OR with change of bases x 3 log 7 log 4 Another method is to take the LOG of both sides Logarithmic Equations Isolate to a single log term Convert to an exponent Solve equation Example log x log x 15 2 log x x 15 2 so 102 x x 15 and 100 x2 15x and 0 x2 15x 100 So 0 x 20 x 5 so x 20 or 5 That s All Folks