Question 706676: HELP (1/3)^x + 6 = (81)^–x
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! What you posted meant:
This is a difficult problem to solve.
But I suspect that what you meant was:
If I am correct, then please put parentheses around multiple-term exponents like x+6. Clearly posted problems get quicker responses. (Also use parentheses around multiple term numerators, denominators, function arguments (like sin(2x-45)), etc. IOW, use parentheses generously to ensure that there is only one correct way to interpret what you post. Tutors more much more likely to respond to problems that are clear.
To solve
we can use logarithms. Logarithms are the usual way to solve equations where the variable is in exponents. But sometimes there is an easier, more exact way to solve these equations. If you can rewrite the equation so that each side of the equation is a power of the same number, then the solution is much easier than with logarithms. So it is worth a little effort to see if this can be done.
Right now the equation says that powers of different numbers, 1/3 and 81, are equal. Is it possible to...- rewrite 1/3 as a power of 81?
- rewrite 81 as a power of 1/3?
- rewrite both 1/3 and 81 as a power of some third number?
If any of these are possible then we have an easy solution in front of us.
If you are clever and good with exponents then you will see how all three of the above being possible. If you cannot see this immediately, then the third option above, changing both numbers into powers of some third number, can be the easiest to figure out. So the question is: 1/3 and 81 can both be rewritten as powers of what number? With some real thought about exponents it should not take long to get the answer: 3!  and  . Replacing 1/3 and 81 with these powers of 3 gives us:
Our equation now has powers of powers. The rule for exponents in this situation is to multiply the exponents. Multiplying the exponents we get:
The equation is now in the desired form: two powers of the same number, 3, which are equal. The only way these powers of three can be equal is if the exponents are equal, too. So:
Now that the variables are out of the exponents we can solve. Adding x to each side:
Dividing by -3:
P.S. When solving equations like this one, it is always worth a little time to see if you can solve it the way we did here because this way, even with the effort it took to figure out that 1/3 and 81 were both powers of 3, is much easier than using logarithms. (Also, if you use your calculator to find logs then you will most likely be working with decimal approximations. These approximations could lead to answers like x = 2.0000001 or x = 1.9999998 instead of the x = 2 we got for this problem.)
However there are equations where it is not possible to rewrite the equation as powers of the same number that are equal. For these equations you must use logarithms.
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