SOLUTION: Solve Equation 7^(x+1)=43 Answer 0.933 Help is appreciated.

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Question 277816: Solve Equation
7^(x+1)=43

Answer 0.933
Help is appreciated.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
7%5E%28x%2B1%29=43
Solving equations where the variable is in an exponent usually involves using logarithms. If you want an exact answer, then you can use a logarithm of any base. But the "best" base to choose the same as the base for which the variable is part of the exponent. In this case this would be base 7. But if you want a decimal approximation then you should choose a base your calculator "knows" (like base 10 or base e (ln)). I'll do the problem both ways.

Using base 7 logarithms to get the simplest exact answer:
log%287%2C+%287%5E%28x%2B1%29%29%29+=+log%287%2C+%2843%29%29
Now we use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent out in front. This property is precisely the reason we use logarithms on problems like this. It allows us to move the variable out of the exponent where we can "get at it". Using this property on the first logarithm we get:
%28x%2B1%29log%287%2C+%287%29%29+=+log%287%2C+%2843%29%29
By definition, log%287%2C+%287%29%29+=+1. (This is why we chose base 7 logarithms.) So the equation simplifies to:
x%2B1+=+log%287%2C+%2843%29%29
Now we just subtract 1 from each side:
x+=+log%287%2C+%2843%29%29+-+1
This is an exact expression for the solution to your equation.

Using base 10 (or base e) logarithms (because we want a decimal approximation):
log%28%287%5E%28x%2B1%29%29%29+=+log%28%2843%29%29
Using the property to move the exponent out in front:
%28x%2B1%29log%28%287%29%29+=+log%28%2843%29%29
Dividing both sides by log(7):
x%2B1+=+log%28%2843%29%29%2Flog%28%287%29%29
Subtracting 1 from each side:
x+=+log%28%2843%29%29%2Flog%28%287%29%29+-+1
Now we can get out our calculators and
  1. Find the two logarithms.
  2. Divide the two logarithms
  3. Subtract 1 from the result of the division.

This should result in something very close to the 0.933.

P.S. If you find the exact answer then decide that you want a decimal, then you can take
x+=+log%287%2C+%2843%29%29+-+1
and use the base conversion formula, log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29, to convert the base 7 logarithm into base 10 (or base e) logarithms.

P.P.S. If you use base e instead of base 10 logarithms to find a decimal approximation, you end up with the same answer! (Try it an see.)