Solve for x

a) 9^(1/2x^2-1)+1=4

b) 5^(2x-1)=2^(1-x)

c) log_base 2_(x+4)+log(x-2)=4

d) log(7x-12)=2logx

 Apr 23, 2011 Equations Involving Logarithms by: Staff ------------------------------------------------- Part II c 5.35, -0.24999 5.36, -0.24715 5.37, -0.24432 5.38, -0.2415 5.39, -0.23868 5.4, -0.23586 5.41, -0.23305 5.42, -0.23025 5.43, -0.22745 5.44, -0.22465 5.45, -0.22187 5.46, -0.21908 5.47, -0.21631 5.48, -0.21353 5.49, -0.21077 5.5, -0.208 5.51, -0.20525 5.52, -0.2025 5.53, -0.19975 5.54, -0.19701 5.55, -0.19427 5.56, -0.19154 5.57, -0.18881 5.58, -0.18609 5.59, -0.18337 5.6, -0.18066 5.61, -0.17796 5.62, -0.17525 5.63, -0.17256 5.64, -0.16987 5.65, -0.16718 5.66, -0.1645 5.67, -0.16182 5.68, -0.15915 5.69, -0.15648 5.7, -0.15381 5.71, -0.15115 5.72, -0.1485 5.73, -0.14585 5.74, -0.14321 5.75, -0.14057 5.76, -0.13793 5.77, -0.1353 5.78, -0.13267 5.79, -0.13005 5.8, -0.12743 5.81, -0.12482 5.82, -0.12221 5.83, -0.11961 5.84, -0.11701 5.85, -0.11442 5.86, -0.11183 5.87, -0.10924 5.88, -0.10666 5.89, -0.10408 5.9, -0.10151 5.91, -0.09894 5.92, -0.09637 5.93, -0.09381 5.94, -0.09126 5.95, -0.08871 5.96, -0.08616 5.97, -0.08362 5.98, -0.08108 5.99, -0.07854 6, -0.07601 6.01, -0.07349 6.02, -0.07096 6.03, -0.06845 6.04, -0.06593 6.05, -0.06342 6.06, -0.06092 6.07, -0.05841 6.08, -0.05592 6.09, -0.05342 6.1, -0.05093 6.11, -0.04845 6.12, -0.04597 6.13, -0.04349 6.14, -0.04101 6.15, -0.03854 6.16, -0.03608 6.17, -0.03362 6.18, -0.03116 6.19, -0.0287 6.2, -0.02625 6.21, -0.02381 6.22, -0.02136 6.23, -0.01893 6.24, -0.01649 6.25, -0.01406 6.26, -0.01163 6.27, -0.00921 6.28, -0.00679 6.29, -0.00437 6.3, -0.00196 6.31, 0.00045 6.32, 0.002855 6.33, 0.005256 6.34, 0.007654 6.35, 0.010048 6.36, 0.012439 6.37, 0.014825 6.38, 0.017209 6.39, 0.019588 6.4, 0.021964 6.41, 0.024337 6.42, 0.026706 6.43, 0.029071 6.44, 0.031433 6.45, 0.033791 6.46, 0.036146 6.47, 0.038497 6.48, 0.040845 6.49, 0.043189 6.5, 0.04553 6.51, 0.047867 6.52, 0.050201 6.53, 0.052532 6.54, 0.054859 6.55, 0.057182 6.56, 0.059503 6.57, 0.06182 6.58, 0.064133 6.59, 0.066443 6.6, 0.06875 6.61, 0.071054 6.62, 0.073354 6.63, 0.075651 6.64, 0.077944 6.65, 0.080234 6.66, 0.082521 6.67, 0.084805 6.68, 0.087086 6.69, 0.089363 6.7, 0.091637 6.71, 0.093907 6.72, 0.096175 6.73, 0.098439 6.74, 0.1007 6.75, 0.102958 6.76, 0.105213 6.77, 0.107465 6.78, 0.109713 6.79, 0.111958 6.8, 0.114201 6.81, 0.11644 6.82, 0.118676 6.83, 0.120908 6.84, 0.123138 6.85, 0.125365 6.86, 0.127588 6.87, 0.129809 6.88, 0.132026 6.89, 0.134241 6.9, 0.136452 6.91, 0.138661 6.92, 0.140866 6.93, 0.143068 6.94, 0.145268 6.95, 0.147464 6.96, 0.149658 6.97, 0.151848 6.98, 0.154035 6.99, 0.15622 7, 0.158402 7.01, 0.16058 7.02, 0.162756 7.03, 0.164929 7.04, 0.167099 7.05, 0.169266 7.06, 0.17143 7.07, 0.173591 7.08, 0.17575 7.09, 0.177905 7.1, 0.180058 7.11, 0.182208

 Apr 23, 2011 Equations Involving Logarithms by: Staff ------------------------------------------------- Part II b 3.75, -0.80277 3.76, -0.79843 3.77, -0.79411 3.78, -0.78981 3.79, -0.78552 3.8, -0.78125 3.81, -0.777 3.82, -0.77276 3.83, -0.76854 3.84, -0.76433 3.85, -0.76014 3.86, -0.75596 3.87, -0.75179 3.88, -0.74765 3.89, -0.74351 3.9, -0.73939 3.91, -0.73529 3.92, -0.7312 3.93, -0.72712 3.94, -0.72306 3.95, -0.71901 3.96, -0.71498 3.97, -0.71095 3.98, -0.70695 3.99, -0.70295 4, -0.69897 4.01, -0.695 4.02, -0.69105 4.03, -0.6871 4.04, -0.68317 4.05, -0.67926 4.06, -0.67535 4.07, -0.67146 4.08, -0.66758 4.09, -0.66371 4.1, -0.65986 4.11, -0.65602 4.12, -0.65218 4.13, -0.64837 4.14, -0.64456 4.15, -0.64076 4.16, -0.63698 4.17, -0.6332 4.18, -0.62944 4.19, -0.62569 4.2, -0.62195 4.21, -0.61823 4.22, -0.61451 4.23, -0.6108 4.24, -0.60711 4.25, -0.60342 4.26, -0.59975 4.27, -0.59609 4.28, -0.59243 4.29, -0.58879 4.3, -0.58516 4.31, -0.58154 4.32, -0.57793 4.33, -0.57433 4.34, -0.57074 4.35, -0.56716 4.36, -0.56359 4.37, -0.56002 4.38, -0.55647 4.39, -0.55293 4.4, -0.5494 4.41, -0.54588 4.42, -0.54236 4.43, -0.53886 4.44, -0.53537 4.45, -0.53188 4.46, -0.52841 4.47, -0.52494 4.48, -0.52148 4.49, -0.51804 4.5, -0.5146 4.51, -0.51117 4.52, -0.50775 4.53, -0.50433 4.54, -0.50093 4.55, -0.49754 4.56, -0.49415 4.57, -0.49077 4.58, -0.4874 4.59, -0.48404 4.6, -0.48069 4.61, -0.47735 4.62, -0.47401 4.63, -0.47068 4.64, -0.46736 4.65, -0.46405 4.66, -0.46075 4.67, -0.45746 4.68, -0.45417 4.69, -0.45089 4.7, -0.44762 4.71, -0.44436 4.72, -0.4411 4.73, -0.43786 4.74, -0.43462 4.75, -0.43138 4.76, -0.42816 4.77, -0.42494 4.78, -0.42173 4.79, -0.41853 4.8, -0.41534 4.81, -0.41215 4.82, -0.40897 4.83, -0.4058 4.84, -0.40264 4.85, -0.39948 4.86, -0.39633 4.87, -0.39318 4.88, -0.39005 4.89, -0.38692 4.9, -0.3838 4.91, -0.38068 4.92, -0.37757 4.93, -0.37447 4.94, -0.37138 4.95, -0.36829 4.96, -0.36521 4.97, -0.36214 4.98, -0.35907 4.99, -0.35601 5, -0.35295 5.01, -0.34991 5.02, -0.34687 5.03, -0.34383 5.04, -0.3408 5.05, -0.33778 5.06, -0.33477 5.07, -0.33176 5.08, -0.32876 5.09, -0.32576 5.1, -0.32277 5.11, -0.31979 5.12, -0.31681 5.13, -0.31384 5.14, -0.31088 5.15, -0.30792 5.16, -0.30497 5.17, -0.30202 5.18, -0.29908 5.19, -0.29614 5.2, -0.29322 5.21, -0.29029 5.22, -0.28738 5.23, -0.28447 5.24, -0.28156 5.25, -0.27866 5.26, -0.27577 5.27, -0.27288 5.28, -0.27 5.29, -0.26713 5.3, -0.26426 5.31, -0.26139 5.32, -0.25853 5.33, -0.25568 5.34, -0.25283 -------------------------------------------------

 Apr 23, 2011 Equations Involving Logarithms by: Staff ------------------------------------------------- Part II a Footnote: Part c) Data points used to plot graph (data points were generated using Microsoft Excel) log_base 2_(x+4)+log(x-2)=4 x, f(x) 2.1, -2.39119 2.11, -2.34743 2.12, -2.30729 2.13, -2.27017 2.14, -2.23563 2.15, -2.20332 2.16, -2.17295 2.17, -2.14428 2.18, -2.11712 2.19, -2.09131 2.2, -2.0667 2.21, -2.04319 2.22, -2.02066 2.23, -1.99904 2.24, -1.97824 2.25, -1.9582 2.26, -1.93886 2.27, -1.92017 2.28, -1.90208 2.29, -1.88454 2.3, -1.86753 2.31, -1.851 2.32, -1.83493 2.33, -1.81928 2.34, -1.80404 2.35, -1.78918 2.36, -1.77467 2.37, -1.7605 2.38, -1.74666 2.39, -1.73312 2.4, -1.71987 2.41, -1.70689 2.42, -1.69418 2.43, -1.68171 2.44, -1.66949 2.45, -1.65749 2.46, -1.64571 2.47, -1.63414 2.48, -1.62276 2.49, -1.61159 2.5, -1.60059 2.51, -1.58977 2.52, -1.57912 2.53, -1.56864 2.54, -1.55832 2.55, -1.54814 2.56, -1.53812 2.57, -1.52823 2.58, -1.51848 2.59, -1.50887 2.6, -1.49938 2.61, -1.49002 2.62, -1.48078 2.63, -1.47165 2.64, -1.46264 2.65, -1.45373 2.66, -1.44493 2.67, -1.43624 2.68, -1.42764 2.69, -1.41914 2.7, -1.41074 2.71, -1.40243 2.72, -1.39421 2.73, -1.38607 2.74, -1.37802 2.75, -1.37005 2.76, -1.36216 2.77, -1.35435 2.78, -1.34662 2.79, -1.33896 2.8, -1.33138 2.81, -1.32386 2.82, -1.31641 2.83, -1.30904 2.84, -1.30172 2.85, -1.29448 2.86, -1.28729 2.87, -1.28017 2.88, -1.27311 2.89, -1.26611 2.9, -1.25916 2.91, -1.25227 2.92, -1.24544 2.93, -1.23866 2.94, -1.23194 2.95, -1.22526 2.96, -1.21864 2.97, -1.21207 2.98, -1.20555 2.99, -1.19907 3, -1.19265 3.01, -1.18626 3.02, -1.17993 3.03, -1.17364 3.04, -1.16739 3.05, -1.16119 3.06, -1.15503 3.07, -1.14891 3.08, -1.14283 3.09, -1.13679 3.1, -1.13079 3.11, -1.12483 3.12, -1.1189 3.13, -1.11302 3.14, -1.10717 3.15, -1.10136 3.16, -1.09558 3.17, -1.08984 3.18, -1.08413 3.19, -1.07846 3.2, -1.07282 3.21, -1.06722 3.22, -1.06164 3.23, -1.0561 3.24, -1.05059 3.25, -1.04511 3.26, -1.03966 3.27, -1.03424 3.28, -1.02885 3.29, -1.02349 3.3, -1.01816 3.31, -1.01286 3.32, -1.00758 3.33, -1.00234 3.34, -0.99712 3.35, -0.99192 3.36, -0.98676 3.37, -0.98161 3.38, -0.9765 3.39, -0.97141 3.4, -0.96635 3.41, -0.96131 3.42, -0.95629 3.43, -0.9513 3.44, -0.94633 3.45, -0.94139 3.46, -0.93647 3.47, -0.93157 3.48, -0.9267 3.49, -0.92185 3.5, -0.91702 3.51, -0.91221 3.52, -0.90742 3.53, -0.90266 3.54, -0.89791 3.55, -0.89319 3.56, -0.88849 3.57, -0.88381 3.58, -0.87915 3.59, -0.8745 3.6, -0.86988 3.61, -0.86528 3.62, -0.86069 3.63, -0.85613 3.64, -0.85158 3.65, -0.84706 3.66, -0.84255 3.67, -0.83806 3.68, -0.83358 3.69, -0.82913 3.7, -0.82469 3.71, -0.82027 3.72, -0.81587 3.73, -0.81149 3.74, -0.80712 -------------------------------------------------

 Apr 23, 2011 Equations Involving Logarithms by: Staff Part I The question: Solve for x a) 9^(1/2x^2-1)+1=4 b) 5^(2x-1)=2^(1-x) c) log_base 2_(x+4)+log(x-2)=4 d) log(7x-12)=2logx The answer: a) 9^(1/2x^2 - 1)+1=4 9^(1/2x^2 - 1) + 1 - 1 = 4 - 1 9^(1/2x^2 - 1) + 0 = 4 - 1 9^(1/2x^2 - 1) = 4 – 1 9^(1/2x^2 - 1) = 3 9^(1/2x^2 - 1) = 9^(1/2) (1/2x^2 - 1) = (1/2) 1/2x^2 - 1 = 1/2 x^2 - 2 = 1 x^2 = 3 sqrt(x^2) = sqrt(3) x = ±sqrt(3) ---------------------------------------------- b) 5^(2x-1)=2^(1-x) 5^(2x-1)=2^(1-x) log5^(2x-1)=log2^(1-x) (2x-1)*log5=(1-x)*log2 2x*log5 - log5 =1*log2 - x*log2 2x*log5 - log5 =log2 - x*log2 2x*log5 - log5 + log5 =log2 - x*log2 + log5 2x*log5 + 0 =log2 - x*log2 + log5 2x*log5 =log2 - x*log2 + log5 2x*log5 + x*log2 = log2 - x*log2 + x*log2 + log5 2x*log5 + x*log2 = log2 + 0 + log5 2x*log5 + x*log2 = log2 + log5 x*(2*log5 + log2) = log2 + log5 x*(2*log5 + log2) / (2*log5 + log2) = (log2 + log5) / (2*log5 + log2) x*1 = (log2 + log5) / (2*log5 + log2) x = (log5 + log2) / (2*log5 + log2) x = 0.588592 --------------------------------------------------------- c) log_base 2_(x+4)+log(x-2)=4 log_base 2_(x+4)+log(x-2)=4 x = 6.31 using a graphical solution log_base 2_(6.31+4)+log(6.31-2)=4 log_base 2_(10.31)+log(4.31)=4 A graphical solution was used x = 6.31 Click the following link to view the graph. Press the Backspace key to return to this page: http://www.solving-math-problems.com/images/graphical-solution-logarithm.png (the table of values used to plot the graph are shown in the footnote) --------------------------------------------------------- d) log(7x - 12)=2logx log(7x - 12)=2logx log(7x - 12)=log(x^2) 7x - 12 = x^2 7x – 7x -12 + 12 = x^2 - 7x + 12 0 + 0 = x^2 - 7x + 12 0 = x^2 – 7x + 12 x^2 - 7x + 12 = 0 (x - 3)*(x - 4) = 0 (x - 3) = 0 x - 3 = 0 x - 3 + 3 = 0 + 3 x - 3 + 3 = 3 x + 0 = 3 x = 3 (x - 4) = 0 x - 4 = 0 x - 4 + 4 = 0 + 4 x - 4 + 4 = 4 x + 0 = 4 x = 4 x = 3 and x = 4 Check the solutions for validity: a) 9^(1/2x^2-1)+1=4 x = ±sqrt(3) 9^(1/2x^2 - 1)+1=4 9^(1/2 * sqrt(3)^2 - 1)+1=4 9^(1/2 * 3 - 1)+1=4 9^(3/2 - 1)+1=4 9^(.5)+1=4 3 + 1 = 4, solution valid --------------------------------------- b) 5^(2x-1)=2^(1-x) x = 0.588592 5^(2x-1)=2^(1-x) 5^(2*0.588592-1)=2^(1-0.588592) 1.32998 = 1.32998, solution valid --------------------------------------- c) log_base 2_(x+4)+log(x-2)=4 x = 6.31 log_base 2_(x+4)+log(x-2)=4 log_base 2_(6.31+4)+log(6.31-2)=4 4.000 = 4, solution valid --------------------------------------- d) log(7x - 12)=2logx x = 3 x = 4 log(7x - 12)=2logx for x = 3 log(7*3 - 12)=2*log(3) 0.954243 = 0.954243, solution valid for x = 3 log(7*4 - 12)=2*log(4) 1.20412 = 1.20412, solution valid Thanks for writing. Staff www.solving-math-problems.com -------------------------------------------------