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LOGS EQUAL THE Logarithmic Function The inverse of an exponential function is a logarithmic function. x = log a y read: “x equals log base a of y” We can convert exponential equations to logarithmic equations and vice versa, using this: Logarithmic form: y = logb x Exponential Form: by = x. The log to the base “b” of “x” is the exponent to which “b” must be raised to obtain “x” y = log2 x y = log b x x=2 y x=by Convert to exponential form 1) ylog 5 3 2) 3) 2log a 7 alog bd 3 5 y 2 a 7 b d a Convert to logarithmic form 4) 5) 6) 10 1000 3 2 8 x y 1 4 3log 1000 10 xlog 8 2 1log 4 y Now that we can convert between the two forms we can simplify logarithmic expressions. Without a Calculator! Simplify “What is the exponent of that gives you 32?” 1) log2 32=x 2x = 32 x=5 “What is the exponent of 3 that gives you 27?” 2) log3 27=x 3x = 27 x=3 3) log4 2=x 4x = 2 x = 1/2 4) log3 1=x 3x = 1 x=0 Evaluate 1 5) log 6 x 36 We can also use these two forms to help us solve for an inverse. The steps for finding an inverse are the same as before. Easy as 1, 2, 3… 1-Rewrite 2-Switch x and y 3-Solve for y Example: Find the inverse y 8 x Isolate the power. 1. rewrite (no-need) 2. Switch x and y 3. Solve for y Rewrite in log form : y 8x x 8y x 8y x 8 log8 x y y y log8 x is the inverse of y 8 x Find the inverse of the following exponential functions… f(x) = 2x f(x) = 2x+1 f(x) = 3x- 1 f-1(x) = log2x f-1(x) = log2x f-1(x) = log3(x + 1 Find the inverse of the log functions. y log 2 x 4 1. Switch x and y 2. Isolate the log 3. Rewrite in exponential form. 4. Solve for y. Find the inverse for the following: 1. y = log3x 2. y = log(2x) 3. y = log5x – 3 Info about Logarithms Common logarithms – have a base of 10, but we do not write the base Ex: log10x = log x Natural logarithms – have a base of e, logex = ln x