a) Simplify 2ln(2x+1) - 10 = 0 b) Simplify 3^(x)*e^(4x) = e^(7)

a) To answer this question, one must be familiar with laws of logs, more sprecifically the rules when applied to the natural log of x, and exponentials (e). 2ln(2x+1) - 10 = 0 Step 1) 2ln(2x+1) = 10 Step 2) ln(2x+1) = 10/2 = 5 Step 3) using the fact that e^(ln(x)) = x, e^(ln(2x+1)) = e^(5) = 2x + 1 Step 4) to find x on its own, we simply rearrange this equation to give x = (e^(5)-1)/2 which is the final answer.                                     b) This question again requires the knowledge of the laws of logs, specifically the natural log of x, and also the rule regarding division of exponential functions. 3^(x)*e^(4x) = e^(7) Step 1) ln(3^(x)*e^(4x)) = ln(e^(7)) = 7 Step 2) ln(3^(x)) + ln(e^(4x)) = 7 Step 3) using 2 different laws of logs, (lna^b = blna) and (lne^(a) = a), xln3 = 7 - 4x Step 4) simple rearrangment gives xln3 + 4x = 7 Step 5) Factorising gives x(ln3 + 4) = 7 and therefore x = 7 / (ln3 + 4)