Question 1200017: Solve the following exponential equations
a) 3^x=5(2^x)
b) 9^(x-1)*3^x=27 Found 3 solutions by ikleyn, Theo, MathTherapy:Answer by ikleyn(52787) (Show Source):
You can put this solution on YOUR website! .
Solve the following exponential equations
a) 3^x=5(2^x)
b) 9^(x-1)*3^x=27
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(a) = .
Take logarithm base 10 of both sides. Use properties of logarithms. You will get
x*log(3) = log(5) + x*log(2)
x*log(3) - x*log(2) = log(5)
x*(log(3) - log(2)) = log(5)
x = = 3.969 (rounded). ANSWER
(b) =
It is the same as
= = = =
The base is 3 in both sides of the equation - hence, the degrees are identical
3x-2 = 3
3x = 3+2 = 5
x = . ANSWER
divide both sides of the equation by 2^x to get:
3^x/2^x = 5
this becomes:
(3/2)^x = 5
take the log of both sides of the equation to getL
log((3/2)^x) = log(5)
this becomes:
x * log(3/2) = log(5)
solve for x to get:
x = log(5)/log(3/2) = 3.969362296.
that's our solution.
confirm by replacing x with that in the original equation to get:
3^x = 5 * 2^x becomes 78.31899737 = 78.31899737, confirming the the solution is good.
b) 9^(x-1)*3^x=27
9^(x-1) is equal to 3^2)^(x-1) which is equal to 3^(2*(x-1) which is equal to 3^(2x-2)
your equation becomes:
3^(2x-2) * 3^x = 27
this becomes 3^(2x-2+x) = 27 which becomes 3^(3x-2) = 27
take the log of both sides of the equation to get:
log(3^(3x-2)) = log(27) which becomes (3x-2)*log(3) = log(27)
divide both sides of the equation by log(3) to get:
3x-2 = log(27)/log(3)
add 2 to both sides of the equation and then divide both sides of the equation by 3 to get:
x = (log(27)/log(3)+2)/3
solve for x to get:
x = 1.666666666..... which is equal to 1 + 2/3 which is equal to 5/3.
replace x with that in the original equation to get:
9^(x-1)*3^x=27 becomes 9^(5/3 - 1) * 3^(5/3) = 27 which becomes 27 = 27, confirming the soluton is correct.
your solution is that x = 5/3 or x = 1.6666667.
5/3 is the exact solution whiile x = 1.6666667 is an approximation rounded to the number of decimal places that can be displayed by the calculator.
use of the following properties have been made of.
log(b^x) = x * log(b)
a^x * a^y = a^(x + y)
(a^2)^y = a^(2y)
