SOLUTION: How would you solve this equation for a? 2^a+3 = 3^2a-1 Thanks

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Question 970559: How would you solve this equation for a? 2^a+3 = 3^2a-1
Thanks
Found 2 solutions by Theo, ankor@dixie-net.com:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
i would solve this as follows:

start with 2^(a+3) = 3^(2a-1)

take the log of both sides of the equation to get:

log(2^(a+3)) = log(3^(2a-1)

since log(x^a) = a*log(x), your equation of log(2^(a+3)) = log(3^(2a-1)
becomes:

(a+3)*log(2) = (2a-1)*log(3)

divide both sides of this equation by log(2) to get:

a+3 = (2a-1)*log(3)/log(2)

to simplify how this looks, we will temporarily let b = log(3)/log(2).

the equation of a+3 = (2a-1)*log(3)/log(2) becomes a+3 = (2a-1)*b

perform the multiplication on the right side of the equation to get:

a+3 = 2ab-b

subtract a from both sides of this equation and add b to both sides of this eqaution to get:

b+3 = 2ab-a

factor out the a on the right side of this equation to get:

b+3 = (2b-1)*a

divide both sides of this eqaution by (2b-1) to get:

(b+3)/(2b-1) = a

we had previously let b = log(3)/log(2).

now we replace b with log(3)/log(2) to get:

(b+3)/(2b-1) = a becomes:

(log(3)/log(2)+3)/(2*log(3)/log(2)-1) = a

use your calculator to solve for a to get a = 2.112958972.

that's your solution.

replace a in the original equation with 2.112958972 and you get:

log(2^(a+3)) = log(3^(2a-1) becomes:

log(2^(2.112958972+3) = log(3^(2*2.112958972-1) which becomes:

34.60620827 = 34.60620827 which is true.

this confirms the solution is good.

the solution is a = 2.112958972

Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website!
solve this equation for a?
Use nat logs

The log equiv of exponents

=
find ln
= 1.58496
a + 3 = 1.58496(2a-1)
a + 3 = 3.1699a - 1.58496
3 + 1.58496 = 3.1699a - a
4.58496 = 2.1699a
a =
a = 2.113
:
:
Checks out pretty close on a calc, enter