SOLUTION: 5^(x-1)=12^x (estimate to 4 places)

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Question 261173: 5^(x-1)=12^x (estimate to 4 places)

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
5%5E%28x-1%29=12%5Ex
Solving equations where the variable is in an exponent often involves the use of logarithms. Since the base's are different (and not powers of the same number), we can use any base logarithm as long as our calculator can figure them out (usually base 10 or base e (ln) logarithms). Since base 10 logarithms are probably the most common to calculators I will use them.

We start by finding the logarithm of each side:
log%28%285%5E%28x-1%29%29%29+=+log%28%2812%5Ex%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 in the argument out in front. (This property is the reason we use logarithms. It allows us the get the variable out of the exponents!)
%28x-1%29log%28%285%29%29+=+x%2Alog%28%2812%29%29
Using the Distributive Property on the left side we get:
x%2Alog%28%285%29%29+-+log%28%285%29%29+=+x%2Alog%28%2812%29%29
Now we solve for x. We start by getting the x's on the same side of the equation. I'll subtract xlog%28%285%29%29 from each side:
-log%28%285%29%29+=+x%2Alog%28%2812%29%29+-+xlog%28%285%29%29
Next we factor out x on the right side:
-log%28%285%29%29+=+x%2A%28log%28%2812%29%29+-+log%28%285%29%29%29
Next we divide both sides by %28log%28%2812%29%29+-+log%28%285%29%29%29
%28-log%28%285%29%29%29%2F%28log%28%2812%29%29+-+log%28%285%29%29%29+=+x
And last of all, we turn to our calculators to find these logarithms and calculate the solution.
%28-0.6989700043360188%29%2F%281.0791812460476248+-+0.6989700043360188%29+=+x
%28-0.6989700043360188%29%2F0.3802112417116060+=+x
-1.8383727982093555+=+x
Rounded to 4 places this is:
-1.8384+=+x

Note that if you replace all the log's in the above equation with ln's the numbers will look different up to the last step. And then, when you divide the last two number, you end up with the same answer as we did with base 10 logarithms!