Study Notes Exponents/Logarithms Math 30 Pure ...

Study Notes Exponents/Logarithms Math 30 Pure1. A logarithm is the same thing as an exponent.log 8 ( 64) = 2log 4 ( 64) = 3log 2 ( 64) = 6log 10 ( 64) = LOG( 64)= 1.8068 2 = 644 3 = 642 6 = 6410 1.806 = 642. The equivalence of logarithmic form and exponential form.M =log b ( N)is equivalent tob M = N3. Important alternate forms for evaluating expressions and solving equations.5 = log 2 ( 2)= log 3 ( 3)= log 7 ( 7)= log 9 ( 9)4. The laws of logarithms, to any base b.product law :quotient law:power law :log b ( N⋅M) = log b ( N) + log b ( M)⎛⎜⎝⎛⎝log bNMlog b M nchange of base : log b ( N)⎞⎟⎠⎞⎠= logb( N) − log b ( M)= n⋅log b ( M)=log a ( N)log a ( B)5. An important example of a logarithmic equation.log 3 ( x − 1) − log 3 ( x)= 2becomes log 3 ( x − 1) − log 3 ( x)= log 3 3 2⎛⎝⎞⎠6. When solving 32 3x−2= 64 switch to base 2 since 32 = 2 5 and 64 = 2 6 .When solving 32 3x−2= 65 take logarithms to base 10 of each side.7. y = b x and y = log b ( x)are inverse functions.

Study Notes Exponents/Logarithms Math 30 Pure8. The exponential function , y = a⋅b x + c .y intercept is a + cx intercept is the solution to 0 =a⋅b x+ chorizontal asymptote is y = cdomain is x all realsrange is y > c9. The logarithmic function , y = log a ( x + c).y intercept is y = log a ( 0 + c)x intercept is the solution to 0 = log a ( x + c)vertical asymptote is x = -cdomain is x> −crange is y all reals710. Growth functions : starting at 500 then doubling in 7 years : y = 500⋅2starting at 700 then increasing at 3% per year : y = 700⋅1.03 xDecay functions : starting at 300 with a half-life of 8 years : y = 300⋅0.5⎛⎜⎜⎝⎛⎜⎜⎝x⎞ ⎟⎟⎠⎛⎝x⎟⎞8⎟⎠⎛⎝starting at 500 then decreasing at 8% per year : y = 500⋅0.92 xstarting at 200 then decreasing by 1 3⎞⎠⎞⎠⎛⎝= 3 − 1 per year : y = 200⋅3 − x⎞⎠