Answer:
x = 0.1116 (rounded to 4 decimal places)
Step-by-step explanation:
We need to isolate "e" first, so we do:
[tex]4e^{2x}=5\\e^{2x}=\frac{5}{4}\\e^{2x}=1.25[/tex]
Solving these types of equations requires us to take the Natural Logarith (Ln) of both sides, so we have:
[tex]e^{2x}=1.25\\Ln(e^{2x})=Ln(1.25)[/tex]
We can use the property of logarithms shown below to further simplify:
[tex]Ln(a^b)=bLn(a)[/tex]
So, we have:
[tex]Ln(e^{2x})=Ln(1.25)\\(2x)Ln(e)=Ln(1.25)[/tex]
We know Ln(e) = 1, thus now, we can replace it and solve for x:
[tex](2x)Ln(e)=Ln(1.25)\\(2x)(1)=Ln(1.25)\\2x=Ln(1.25)\\2x=0.2231\\x=\frac{0.2231}{2}\\x=0.1116[/tex]
So
x = 0.1116 (rounded to 4 decimal places)
