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Logarithms The “I’m going to lie to you a bit” version Example • Every year I double my money $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 $1 Year 1 $1 $1 Year 2 Year 3 Example • If I know what time it is and want to know how much money I have $1 t a=1(2 ) a=# of $ t=# of yrs $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 $1 Year 1 $1 $1 Year 2 Year 3 Example • What if I know money and want to know time? $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 $1 Year 1 $1 $1 Year 2 Year 3 Example • How long would it take me to get $1,000,000? $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 $1 Year 1 $1 $1 Year 2 Year 3 Two sequences +1 +1 +1 +1 +1 +1 +1 +1 Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 2 4 8 16 32 54 128 *2 *2 *2 *2 *2 *2 *2 *2 Exponential Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 2 4 8 16 32 54 128 *2 a=2t Logarithmic Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 2 4 8 16 32 54 128 *2 t=log2a Exponential Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 3 9 27 81 243 729 2187 *3 a=3t Logarithmic Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 3 9 27 81 243 729 2187 *3 t=log3a Exponential The special base 10 Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 10 100 1000 10000 100000 100000 10000000 *10 a=10t Logarithmic The special base 10 Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 10 100 1000 10000 100000 100000 10000000 *10 t=log10a t=log(a) Exponential The special base e Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 e e2 e3 e4 e5 e6 e7 *e a=et Logarithmic The special base e Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 e e2 e3 e4 e5 e6 e7 *e t=logea t=ln(a) Logarithmic The special base e e≈2.7182818284 This number makes calculus easier Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 e e2 e3 e4 e5 e6 e7 *e t=logea t=ln(a) Meanings • a=2t I know t. a is the result I get from raising 2 to the t power. • t=log2(a) I know a. t is the power I need to raise 2 to to get a. Example • a=23 a is the result I get from raising 2 to the power 3. That result is 8. a=8. • t=log2(8) t is the power I need to raise 2 to to get 8. Since 23=8, t=3. Exponential a=2t Years (t) -3 -2 1 0 1 2 3 4 Dollars (a) 1/8 ¼ ½ 1 2 4 8 16 *2 No matter what power I use, the result is always positive Exponential a=2t Years (t) -3 -2 1 0 1 2 3 4 Dollars (a) 1/8 ¼ ½ 1 2 4 8 16 *2 There is no power that can get me a negative number Logarithmic t=log2a Years (t) -3 -2 -1 0 1 2 3 4 Dollars (a) 1/8 ¼ ½ 1 2 4 8 16 *2 There is no power that can get me a negative number Logarithmic t=log2a Years (t) -3 -2 -1 0 1 2 3 4 Dollars (a) 1/8 ¼ ½ 1 2 4 8 16 *2 I can only find the powers of positive numbers Logarithmic t=log2a Years (t) -3 -2 -1 0 1 2 3 4 Dollars (a) 1/8 ¼ ½ 1 2 4 8 16 *2 I can only take the log of positive numbers Logarithmic t=log2a Years (t) -3 -2 -1 0 1 2 3 4 Dollars (a) 1/8 ¼ ½ 1 2 4 8 16 *2 The domain of this function called “log2” is a>0 Logarithmic t=log2a Years (t) -3 -2 -1 0 1 2 3 4 Dollars (a) 1/8 ¼ ½ 1 2 4 8 16 *2 The domain of any log(whatever) is always whatever>0 Example problem • Find the domain of 2log7(4x-3)+7x-9 Whatever is inside the log has to be >0. I can find an answer whenever 4x-3>0 x>3/4 Find the domain: a) b) c) d) e) x > 5/4 x < 5/4 x > -5/4 x < -5/4 None of the above Find the domain: Whatever is inside the log has to be >0 5-4x>0 5>4x 5/4>x b) x < 5/4 Rewriting equations • • • • • • y=bx 2=3p q+3=79 9=32x+1 7=e4 x+2=102x-1 x=logby p=log32 9=log7(q+3) 2x+1=log39 4=ln(7) or 2x-1=log(x+2) or 4=loge(7) 2x-1=log10(x+2) • The result of the log is the exponent. • The result of the exponent is what goes inside the log. Meanings • a=2t I know t. a is the result I get from raising 2 to the t power. • t=log2(a) I know a. t is the power I need to raise 2 to to get a. What is 2log2(x)? the result I get from raising 2 to the power I need to raise 2 to to get x = x. Meanings • a=2t I know t. a is the result I get from raising 2 to the t power. • t=log2(a) I know a. t is the power I need to raise 2 to to get a. What is log2(2x)? The power that I need to raise 2 to so that I get the result of raising 2 to the x power. =x Rewriting equations version 2 • 9=32x+1 • Taking the log of both sides. Log3(9)=Log3(32x+1) Log39=2x+1 • Exponentiating both sides 3Log3(9)=32x+1 9=32x+1 Rewriting equations version 2 • 3x-7=e2x+1 • Taking the log of both sides. ln(3x-1)=ln(e2x+1) ln(3x-7)=2x+1 • Exponentiating both sides eln(3x-7)=e2x+1 3x-7=e2x+1 Convert the following logarithmic expression into exponential form: y = ln(x+2). a) ey = x+2 b) 10y = x+2 c) e(x+2) = y d) 10 (x+2) = y e) None of the above Convert the following logarithmic expression into exponential form: y = ln(x+2). y = ln(x+2) ey=eln(x+2) ey=x+2 A Properties of logs Exponential Property adding Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 2 4 8 16 32 54 128 multiplying 23+4=2324 Logarithmic Property adding Years (t) 0 1 2 3 4 5 6 7 Dollars (a) 1 2 4 8 16 32 54 128 multiplying Log(8*16)=log(8)+log(16) The basic property of logarithims • Loga(bc)=logab+Logac Example • • • • • Loga(b4) =loga(bbbb) =loga(b)+Loga(bbb) =loga(b)+Loga(b) +Loga(b) +Loga(b) =4Loga(b) The basic properties of logarithims • Loga(bc)=logab+Logac • Loga(bn)=n*logab Example • • • • • • • • x=log832 what is x? Rewrite as an exponential equation 8x=32 Take log2 of both sides Log2(8x)=Log232 xLog2(8)=Log232 x=Log2(32)/Log2(8) x=5/3 Change of base • • • • • • • • x=logay what is x? Rewrite as an exponential equation ax=y Take logc of both sides Logc(ax)=Logcy xLogc(a)=Logcy x=Logc(y)/Logc(a) logay=Logc(y)/Logc(a) Change of base • Using this rule on your calculator • logay=Logc(y)/Logc(a) If you’re looking for the logay use… Log(y)÷Log(a) Or ln(y)÷ln(a) The basic properties of logarithims • Loga(bc)=logab+Logac • Loga(bn)=n*logab • Logab=logc(b)/logc(a) Side effect: you only ever need one log button on your calculator. Logab=log(b)/log(a) Logab=ln(b)/ln(a) Warning: Remember order of operations WRONG log(2ax) =x*log(2a) =x*[log(2)+log(a)] =x log 2 + x log a CORRECT log(2ax) =log(2(ax)) =log(2)+log(ax) =log(2)+x*log(a) What about division? • • • • • • Loga(b/c) =Loga(b(1/c)) =Loga(bc-1) =Loga(b) + Loga(c-1) =Loga(b) + -1*Loga(c) =Loga(b) - Loga(c) The advanced properties of logarithims • Loga(bc)=logab+Logac • Loga(bn)=n*logab • Logab=logc(b)/logc(a) Side effect: you only ever need one log button on your calculator. Logab=log(b)/log(a) Logab=ln(b)/ln(a) Loga(b/c)=logab-Logac What about roots? log a ( n b ) 1/n log a (b ) 1 log a (b) n The advanced properties of logarithims • Loga(bc)=logab+Logac • Loga(bn)=n*logab • Logab=logc(b)/logc(a) Side effect: you only ever need one log button on your calculator. Logab=log(b)/log(a) Logab=ln(b)/ln(a) Loga(b/c)=logab-Logac Loga(n√b̅)=[logab]/n Expand using the properties of logarithms: log(x2y) a) b) c) d) e) log(x) + log(y) log(x)-log(y) 2 log(x)-log(y) 2 log(x)+log(y) None of the above Expand using the properties of logarithms: log(x2y) log(x2y) =log((x2)y) =log(x2)+log(y) =2log(x)+log(y) D