SOLUTION: 8e^2x=136

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Question 613779: 8e^2x=136
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
8e%5E%282x%29=136
First, isolate the base and its exponent by dividing both sides by 8:
e%5E%282x%29=17
Now find the log of each side. While any base of logarithm can be used, it makes the most sense to use base e logs (aka ln) because
  • matching the base of the logarithm to the base of the exponent will lead to a simpler answer
  • your calculator "knows" base e logs. This will make it easy to find a decimal approximation of the answer if desired.

ln%28e%5E%282x%29%29+=+ln%2817%29

Next use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, which allows us to move the exponent of an argument out in front of the logarithm. (It is this property that is the very reason we use logarithms on these equations. Being able to move the exponent, where the variable is, like this the variable is no longer in an exponent. We can then use "regular" algebra to solve for the variable.) Using this property on our equation we get:
2x%2Aln%28e%29+=+ln%2817%29

Since log%28a%2C+%28a%29%29+=+1 for all bases, the ln(e) = 1. (This is why matching the base of the logarithm to the base of the exponent is advantageous.) The left side becomes:
2x+=+ln%2817%29
Now we just divide by 2:
x+=+ln%2817%29%2F2
This is an exact expression for the solution to this problem. If you want/need a decimal approximation you can use your calculator to find ln(17) and then to divide by 2.