SOLUTION: log(base 7)3^(x-1)=2

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Question 419035: log(base 7)3^(x-1)=2
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%287%2C+%283%5E%28x-1%29%29%29=2
Solving equations where the variable is in an exponent usually involves the use of logarithms. Logarithms are used because thay have a property, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, which allows you to move an exponent out in front where we can "get at it" to solve for the variable.

Your equation already has the logarithm in it. So we go straight to using the property to move the exponent:
%28x-1%29log%287%2C+%283%29%29=2
To solve for x we will start by dividing both sides by that logarithm:
x-1=%282%2Flog%287%2C+%283%29%29%29
And last of all, add 1:
x=%282%2Flog%287%2C+%283%29%29%29%2B1
This is an exact expression for the solution. If you need/want a decimal approximation, them use the change of base formula for logarithms, log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29, to convert the base 7 log into an expression of logs whose base your calculator "knows", like base 10 or base e (aka ln). Then use your calculator to find the two logarithms and simplify.