SOLUTION: how do you solve 7= (1/2)e^x+5-3

Question 614637: how do you solve 7= (1/2)e^x+5-3
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
What you posted meant:

Since problems are not usually presented with something as easily simplified as 5-3, I'm assuming the equation is really:

If this is correct, then please put multiple-term exponents in parentheses in the future.

Solving an equation where the variable is in an exponent usually starts with isolating the base and its exponent. Here we need to eliminate the -3 by adding 3:

And then eliminate the 1/2 my multiplying by its reciprocal, 2. (You could also divide by 1/2 but who wants to divide by a fraction?)

The base, e, and its exponent, x+5, are now isolated.

The next step is to use logarithms. Any base of logarithm can be used but there are advantages to choosing certain bases:

Choosing a base for the logarithm that matches the base of the exponent will result in the simplest expression for the answer.
Choosing a base for the logarithm that your calculator "knows", base 10 or base e (aka ln), will result in a less simple expression but one that can be easily converted into a decimal approximation.

We can get both of these advantages in this problem by picking base e logarithms (better known as ln). Finding the ln of each side:

Next we use a property of exponents, , which allows us to move the exponent of the argument of a logarithm out in front of the logarithm. (This property is the very reason we use logarithms on equations like this. Being able to move an exponent, where the variable is, out in front gives us an equation we can now solve. This property applies to all bases of logarithms which is why any base of logarithm can be used.) Using this property on our equation we get:

By definition for all bases and since the base of ln is e, ln(e) = 1. (This is why matching the base of the logarithm to the base of the exponent gives us the simplest expression.) This leaves us with:

Now that the variable is finally out of the exponent, we can solve for x. All that needs to be done is subtract 5 from each side:

This is the simplest possible exact expression for the solution.

If you want/need a decimal approximation just get out your calculator, find the ln(20) and then subtract 5.

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