Solve the exponential equation.

Question

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)

 

Answer 1

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 :)

 

Answer 2

(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 √

Answer 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.