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Solve the exponential equation.

Solve the exponential equation.
 
 
        x - 3
1 }        =  2
   6               3
 
(type a integer or a decimal.  Round to the nearest thousandth as needed. Use a comma to seperate the answers as needed)
 
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3 Answers

TO solve this equation we use logarithms as follows:
 
(1/6)x-3=(2/3)
Take the natural logarithm (can also use log10) of both sides and you get:
ln(1/6)x-3=ln(2/3)
Because of the logarithmic property: ln(A)x=xln(A) we get:
(x-3)ln(1/6)=ln(2/3)
If we divide both sides by ln(1/6) we get:
x-3=[ln(2/3)]/[ln(1/6)]
add 3 to both sides and we get:
x=[ln(2/3)]/[ln(1/6)]+3
which has a decimal expansion as follows:
3.226 (rounded to thousandth as requested)
Hope this helps :)
 
(1/6)^(x-3) = 2/3
 
log((1/6)^(x-3)) = log(2/3)
 
(x-3) log(1/6) = log(2/3)
 
x log(1/6) - 3 log(1/6) = log(2/3)
 
x log(1/6) = 3 log(1/6) + log(2/3)
 
x = 3 + log(2/3)/log(1/6)
 
x ≈ 3.22629438553092 ≈ 3.226
 
check:
 
(1/6)^(3.22629438553092-3) =? 2/3
 
0.666666666666666 = 2/3 √
(1/6) ^( X -3) = 2/3
 
   ( X -3 ) log ( 1/6) = log( 2/3)
 
     X -3 = log ( 2/3) / log ( 1/6)
    
     X = 3 + 0.2263 = 3. 2263
 
    Test :
 
    ( 1/6) ^ ( 0.2263) = 0.66665996
 
       It works.
 
 
 

Comments

 The answer is tested, and it woks.i