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You have provided the equation e^(3x) = 12, and we have to solve this for x.
We first take the log to the base e for both the sides.
=> ln e^(3x) = ln 12
use the exponent rule for algorithm which states that log a^b = b*log a.
=> 3x ln e = ln 12
ln e = 1
=> 3x = ln 12
=> x = (ln 12)/3
=> x = 2.484 / 3
=> x = .8283
Therefore x is equal to (ln 12)/ 3 or 0.8283.
We notice that the unknown is in superscript. To determine the variable, we'll have to take natural logarithms both sides (we'll take natural logarithms instead of decimal logarithms because the exponential functionhas the base = e).
ln [e^(3x)] = ln 12
We'll apply power rule of logarithms:
3x ln e = ln 12
But ln e = 1 and the equation will become:
3x = ln 12
We'll divide by 3:
x = (1/3)*ln 12
We'll apply again the power rule:
x = ln 12^(1/3)
The solution of the equation is:
x = ln 12^(1/3).
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