Solving Exponential and Logarithmic Equation

Welcome! Looking Ahead: *Today: Last new topic *Wednesday: Practice! Thursday: LAST TEST Take out notes for warm up, please. 1) EXPONENTIAL EQUATIONS A) Bases on BOTH sides of the sad sige Last new notes for the unitt *If the loves ore the same Cancel the beses on each side, set the exponents equid to enchanther and save the remaine Exemples EASY 3x - 1 = 7x + 2 - x ^ 2 = - 3x - 4 w = x ^ 2 - 3x - 4 Cunge ose bine for both bases) teammen "If the bases are NOT the same geber bleed follow the stepe above exponertst 3 ^ (x + 5) =(3^ 2 )^ x- 3 ^ 15 = 3 ^ (2x + 2) x + 5 = 2x + 2 3 = x (2 ^ 5) ^ (x - 1) - 2 ^ (5x) 5(x - 1) = 5x 8x - 5 = 5x ^ 2 Warm Up: 1) Condense: 3 * log(x) + 1/2 * log(y) logx³ +logy 3 logxy log(x ^ 3) * sqrt(y) 2) Expand: log. 4a2 log_4(log_4(a ^ 2)) ^ 4 - log_4(b ^ 4) log_4(2) * log_4(a) - 1 3 log 4 | 2 * log_y(a) - 1/3 * log_y(b) B) Exponent on ONE side of the aquel sige 3=9 log_1(q) = 2 Taclate the exponential 2) Change to a legerithin ***You may have to use the change of base formula*** 5) 27 / 5) 1/10 * (5) ^ (n + 1) = 151 7) e ^ (- x + 1) = 6 0.19 ^ (2x - 1) + 4 * 21 log_2(7) = x X = (log(7))/(log(2)) x = 2.81 221?7 5 ^ (x + 4) = 180 o*g_{5} * 180 = x + 4 3.23 = x + 4 x = 0.77 ln(6) = - x + 1 179 = - x + 1 679 = - x (1-0.79 (7n - 4)/(0 ^ (2 * 3) = 17) log17 2243 23 - 2x - 3 1.23-2x x = 2.12 MAIN IDEA When you are Solping to the 9) You have deported $2000s will it take your money to that pyrreterest, congen 1000 3000 2 d = p * (1 + x/n) ^ m i =1000(1+ c5 2 )^ 2 pm 3 = 1.025 ^ (4t) log 2.a25 3 = 2t 445 * 2 + t = 2225 1900 wil it take for your bobncc to nach $25000 A = 2V(x) 1540 A = P * d deg 2500/1540 = (1540e)/1590 1.62 = e ^ (3it) ln(162) = 0.031t 0.48 = 0.031t t = 15, 481 t = 15.63 SOLVE 1 ore than are log on the same side of the Con ALMAVI CHECK FOR AN EXTRANEOUI IOLITION Youn 107 aver after 2 log overline x = log(121) log(A) + log(a) = log_10(Bb) log_3(4x) = log_8(36) 4x = 36 ( x = 9 deg LOG 1) LOGARITHMIC EQUATIONS: Let's go the other way! 1) Mors than one log on the same side of the equal sign Conilema each sitle first 2) Log on BOTH sides of the squal sig Cancel the logs an each side and solve the remaining equation 3) Lug on ONE side of the equal sign lolate the login Change to an exponential and aufve the equation. ***ALWAYS CHECK FOR AN EXTRANEOUS SOLUTION Y NOT have a negative number aftar a lag! ***Negative ans. is OK if value after the log stays (+) 11) log_3(3x - 6) + log_3(2x + 1) ) 3x - 6 = 2x + 1 x - 7 ln\{n ^ 2 - 30\} = sin 5 x ^ 2 - 30 = 6 (x ^ 2 * sqrt(36))/(x = 16) *No nostive inside a log of the Cancol the bigs on thing ALWAVI CHECK FOR AN EXTRANE QUI SOLUTION log_x(a) ^ 2 + log_x(x + 1) = log_x(x + 6) log( sqrt x(x + 2) )=log x (x+6) x²+ x = X + 6 X ^ 2 + X - 6 = 0 (x + 3)(x - 2) = 0 \\ x=-x2 aligned 16) 3 * log(2) - 3i * log(8) - log(a) log(x ^ 4) - log_7(8 ^ 19) = log(x) ln(16) - ln(2) > ln(x) - log_4(10/2) = log_1(x) 3 = x log_x(x ^ 2) = log_y(121) sqrt(x ^ 2) = boxed 121 x = plus/minus 11 x = 11 Condenach ALMAVI CHECK FOR AN EXTRANEOUS OTION log|dx-2)= log_2(x - 2) - log_2(x + 2) x(x - 1) = log_(x -> ∞)((x - 2)/(x + z)) ( r + 2 * (y - 1)/1 = x-2 x+2 .(x+2) (Y + 2YX - 1) = X - 2 x ^ 2 + 2x - 4 + 2 = x - 2 X =0 20) 5 + 2 * log_2(x) = 21; 3 - 5 (2 * log_2(x))/2 = 16/2 log_2(x) = 8 x = 256 2 ^ 8 = x two approaches to #19... what would you do? *19) ln(x) - ln(5) = 0 two approaches to #20... what would you do? 20) 5 + 2 * log_2(x) = 21 n_{e} ^ (x/5) = 0 10-105-0 log_5(48) = log_3(x) ( 48 = x Rewrite in exponentul form. 3-9 log_5(q) = 2 e deg = x/5 1 = x/5 (x-5) x = 5 We did it!!! Logs Done. Now, we get good at it Joke Sheet Practice "What lies down a hundred feet in the air?" ALL WORK ON SEPERATE PAPER: Copy original problem. Graded Classwork... Hand in!