SOLUTION: -2*13^(3x-9)+8=-69 Solve the equation

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Question 814200: -2*13^(3x-9)+8=-69
Solve the equation
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
-2%2A13%5E%283x-9%29%2B8=-69
First, isolate the base and its exponent. Subtracting 8 we get:
-2%2A13%5E%283x-9%29=-77
Dividing by -2:
13%5E%283x-9%29=77%2F2

Next we find the logarithm of each side. Any base of logarithm may be used. However there are advantages to certain bases:
  • Matching the base of the logarithm to the base of the exponent will lead to the simplest possible expression for the solution. In this problem we would use base 13 logs.
  • Choosing a base of logarithm that your calculator "knows" will lead to a solution which can easily be converted to a decimal approximation (if one is desired).
I will show you both of these smarter choices for the base of the logarithm.

Matching the base (by using base 13 logs):
log%2813%2C+%2813%5E%283x-9%29%29%29=log%2813%2C+%2877%2F2%29%29
Next we use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent of the argument out in front of the log on the left. (It is this very property that is the reason we use logarithms on these equations. It allows to move the exponent, where the variable is, to a location we we can "get at it" with "regular" algebra.)
%283x-9%29log%2813%2C+%2813%29%29=log%2813%2C+%2877%2F2%29%29
By definition, log%2813%2C+%2813%29%29=1. (This is why matching the bases results in simpler expressions.) So this becomes:
3x-9=log%2813%2C+%2877%2F2%29%29
Now we solve for x (now that we can "get at it"). Adding 9:
3x=9%2Blog%2813%2C+%2877%2F2%29%29
Dividing by 3 (or multiplying by 1/3):
x=3%2Blog%2813%2C+%2877%2F2%29%29%2F3
This is an exact expression for the solution to the equation. If you want a decimal approximation for this, either:
  • Re-solve the equation using base 10 or base e logs (which your calculator "knows"), or...
  • Use the base conversion formula on the base 13 log to convert it into an expression of base 10 or base e logs.

Using a base my calculator "knows". Most calculators, if they do logarithms at all, will do both base 10 logs, "log", or base e (aka natural logarithms), "ln". Either of these can be used. I will choose base e logs. (The reasoning for each step here is almost exactly the same as what we used with base 13 logs. So I will leave out explanations except to explain the differences.)
ln%2813%5E%283x-9%29%29=ln%2877%2F2%29
%283x-9%29ln%2813%29=ln%2877%2F2%29
Here ln(13) does not simplify. So we use the Distributive Property to multiply:
3x%2Aln%2813%29-9%2Aln%2813%29=ln%2877%2F2%29
Adding 9*ln(13):
3x%2Aln%2813%29=9%2Aln%2813%29%2Bln%2877%2F2%29
Dividing by 3ln(13):
x=%289%2Aln%2813%29%2Bln%2877%2F2%29%29%2F%283%2Aln%2813%29%29
Even though it looks much different, this is also an exact expression for the solution to your equation. It is not as simple as the one we got when we used base 13 logs but it is easier to find it's decimal approximation.