What is the exact value of x in the exponential equation 4.6 + e7x = 42.6?

You sort of have this in the wrong place, but I'll answer it anyway. One of the things you need to do is use brackets so we know exactly what is going on. In this case, I think you mean e^(7x) and that is the way I will solve it.

Step One
Subtract 4.6 from both sides.
e^(7x) = 42.6 - 4.6
e^(7x) = 38.0             Take the ln of both sides.
ln(e^(7x)) = ln(38.0)
ln(e^(7x)) = 3.6376    Take down the 7x
7x * ln(e) = 3.6376     The ln(e) = 1
7x = 3.6376                Divide by 7
x = 3.6376 / 7
x = 0.5196 <<<<<<<<Answer   

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