SOLUTION: find the solutions of the exponential equations so the problem is 7^x/2=5^1-x so far i have came up to here... x/2 log 7=(1-x) log 5 now what?

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: find the solutions of the exponential equations so the problem is 7^x/2=5^1-x so far i have came up to here... x/2 log 7=(1-x) log 5 now what?      Log On


   

Question 615923: find the solutions of the exponential equations
so the problem is 7^x/2=5^1-x
so far i have came up to here...
x/2 log 7=(1-x) log 5
now what?
Found 2 solutions by jsmallt9, josmiceli:
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%2F2%29log%28%287%29%29=%281-x%29log%28%285%29%29
Good start. Now that the x's are out of the exponents we can solve for it. As usual we start by simplifying. The right side can be multiplied out using the Distributive Property:
%28x%2F2%29log%28%287%29%29=log%28%285%29%29-x%2Alog%28%285%29%29
Next, let;s get rid of the fraction by multiplying each side by 2:
x%2Alog%28%287%29%29=2log%28%285%29%29-2x%2Alog%28%285%29%29
Next, gather the variable terms on one side. Adding 2x*log(5) to each side:
x%2Alog%28%287%29%29%2B2x%2Alog%28%285%29%29=2log%28%285%29%29
The terms on the left are not like terms so we cannot add them (which would be the normal next step). What we do in this situation is factor out the variable:
x%28log%28%287%29%29%2B2log%28%285%29%29%29=2log%28%285%29%29
Now we divide by %28log%28%287%29%29%2B2log%28%285%29%29%29:
x=%282log%28%285%29%29%29%2F%28log%28%287%29%29%2B2log%28%285%29%29%29
This is an exact expression for the solution to your equation. If you want/need a decimal approximation, reach for your calculator.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+7%5E%28x%2F2%29+=+5%5E%281-x%29++
+7%5E%28x%2F2%29+=+5%5E1%2A5%5E%28-x%29+
+7%5E%28x%2F2%29+=+5%2F5%5Ex+
Square both sides
+7%5Ex+=++25+%2F+5%5E%282x%29+
+7%5Ex+%2A+5%5E%282x%29+=+25+
+7%5Ex+%2A+5%5Ex+%2A+5%5Ex++=+25+
+%28+7%2A5%2A5+%29%5Ex+=+25+
+175%5Ex+=+25+
+x+%2A+log%28+175+%29+=+log%28+25+%29+
+x+=+log%2825%29+%2F+log%28175%29+
+x+=+1.39794+%2F+2.24304+
+x+=+.62323+
check:
+7%5E%28x%2F2%29+=+5%5E%281-x%29++
+7%5E%28.62323%2F2%29+=+5%5E%281-.62323%29++
+7%5E.31162+=+5%5E.37677+
+1.83379+=+1.83379+
OK