Modulus Functions 2

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Submitted by roamingfree on Fri, 02/05/2021 - 11:40
skills
To complete this question, you need to be familiar with the modulus function and how to eliminate it to solve equations.
Question
Solve the equation below, determining the value of $x$ in terms of $k$ $$ |e^{2x} - 9k| = 5k $$
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The answers are $x = \frac{1}{2}\ln(14k)$ and $x = \frac{1}{2}\ln(4k)$
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To solve the equation $|e^{2x} - 9k| =5k$, we need to consider two equations
  1. The equation $+(e^{2x} -9k) = 5k$
  2. The equation $-(e^{2x} - 9k) = 5k$

Consider the first equation $$\begin{array}{rcl} (e^{2x} -9k) &=& 5k \\ 2x &=& \ln(14k) \\ x &=& \frac{1}{2}\ln(14k) \end{array} $$ The second question yields $$\begin{array}{rcl} -(e^{2x} - 9k) &=& 5k \\ 2x &=& \ln(4k) \\ x &=& \frac{1}{2}\ln(4k) \\ \end{array} $$