Question 421691: 4e^-2x=17
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
Solving an equation where the variable is in an exponent usually involves logarithms. Although we could start with logarithms right now, it makes things easier to isolate the base and its exponent first. By dividing both sides by 4 we get:
Now we use logarithms. Any base of logarithm can be used. However if we choose a base of logarithm- that matches the base of the exponent, we will end up with a simpler expression.
- that our calculator "knows", like base 10 or base e (aka ln), then we end up with an expression that would be easy to convert to a decimal approximation.
Since base e logarithms match both of these criteria for a "smart" choice of logarithm, we will use them:
Next we use a property of logarithms,  , which allows us to move the exponent of the argument out in front of the logarithm (as a coefficient). (It is this very property that is the reason we use logarithms. It allows us to move an exponent, where the variable is, to a location where we can then solve for that variable. And we can use any base of logarithm because this property applies to all bases of logarithm.) Using this property on our equation we get:
By definition ln(e) = 1. (This is why matching the base of the logarithm to the base of the exponent results in a simpler expression.) So now we have:
Now that the variable is out o the exponent we can easily solve for x. Dividing both sides by -2 we get:
This is an exact expression for the solution to the equation. If you need/want a decimal approximation, get out your calculator.
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