Solve each equation. Express irrational solutions in exactform and as a decimal

Question

Solve each equation. Express irrational solutions in exactform and as a decimal rounded to 3 places.

1) 2-x =1.5

2) 2x+1 =51-2x

3) ex+3=x

4) log4x+log4(x-3)=1

Thanks for your help!

Explanation / Answer

1) 2-x = 1.5 => log2[2-x] = log2 [3/2] => -x log2[2] = log2[3] - log2[2] => -x =log2[3] - log2[2] => x = log2[2] - log2[3] = log2[2/3]˜ -.585 2) 2x+1 = 51-2x =>2x21 = 515-2x => 5/2= 2x/5-2x => 5/2 =2x52x => log5/2[5/2] =log5/2[2x52x] => 1 = log5/2[2x] +log5/2[52x] => 1 = x log5/2[2]+ 2x log5/2[5] => 1 = x (log5/2[2] + 2log5/2[5]) => x = 1 / (log5/2[2] + 2 log5/2[5]) => x = 1/ (log5/2[2] + log5/2[52]) => x= 1 / log5/2[2*25] = 1 / log5/2[50] ˜.234 3) ex+3 = x =>ln[ex+3] = ln[x] => x + 3 = xln[] => x - x ln[] = -3 => x(1 - ln[]) = -3=> x = -3 / (1 - ln[]) = -3 / (ln[e] -ln[]) = -3 / ln[e/] = ˜ 20.728 4) log4[x] + log4[x-3] = 1 =>log4[x(x-3)] = 1 => log4[x2 -3x] = 1 => x2 - 3x = 4 => x2 - 3x - 4 =0 => x = (-b ± sqrt[b2 - 4ac]) /(2a), a = 1, b = -3, c = -4 => x = (-(-3) ±sqrt[(-3)2 - 4(1)(-4)]) / (2(1)) => x = (3 ± sqrt[9 + 16]) / 2 => x = (3 ± 5) / 2=> x = 4 or -1 The final answer is 4 because the log function of any base isundefined when x is negative.

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