solve each equation algebraically and check it by substituing into the orignall equation.

Question

50e^0.035x=200

3LN(x-3)+4=5

Method of your choice by solving.

logx^2=6

Logx^4=2

2x-2^-x/2=4

e^x+e^-x/2=4

500/1+25e^.3x=200

Answer 1

50e^(0.035x)=200
e^(.035x) = 4
take ln of both sides
ln(e^(.035x)) = ln4
.035x lne = ln4, but lne = 1
x = ln4 / .035 = appr. 39.6

log x^2 = 6
by definition:
x^2 = 10^6
x = 10^3 = 1000

do logx^4 = 2 the same way

2x-2^-x/2=4 : I have a feeling that is not what you meant, without brackets I cannot tell what the equation is. I think the first term is probably 2^x

e^x+e^-x/2=4

Again, I think you meant:
e^x+e^(-x/2) = 4

let e^(-x/2) = y , where y is a positive real number
then e^(x/2) = 1/y
and e^x = 1/y^2

so 1/y2 + y = 4
1 + y^3 = 4y^2
y^3 - 4y^2 + 1 = 0
I used a program to find
y = .54 or y = 3.94 appr. , there is also a negative root which would not be allowed

so e^(-x/2) = .54 or ..... use the other root
-x/2 = ln.54
x = 1.2324 or ......

last question, way too ambiguous without brackets.