How do you solve 6^(x-3)=3^(4x+1)?

Answer 1

To solve it you need to use a log function. For this problem, I'll use a natural log (ln), but that's just because it only has two letters and is easy to type. Understand, you can use any log for it.

start, run both sides through the ln function

ln (6^(x-3))=ln (3^(4x+1))

the property of logs is you can take out the exponent from the equation like so:

(x-3) ln (6)= (4x-1) ln (3)

next, divide both sides by ln (6) and (4x-1)

(x-3)/(4x-1)=(ln (3))/(ln (6))

now you plug (ln 3)/(ln 6) into your calculator. It becomes about 0.613. Now substitute that into the equation.

(x-3)/(4x-1)= 0.613

multiply 4x-1 to both sides:

(x-3) = 0.613(4x-1)

becomes

x-3 = 2.452x - 0.613

now solve for x

1.452x = -3.613

and divide

x = 1.452/-3.613

and your solution is:

x = -.40188209

Keep in mind this is only an approximation of "x".

Answer 2

This equation can be solved easily by using the logarithmic function:
6^(x-3)=3^(4x+1) is equivalent with
(x-3)*log(6)=(4x+1)*log(3)
The answer is:
x= -2.487385

Answer 3

(x-3)log6=(4x+1)log3

let log6=a and log3=b

a(x-3)=b(4x+1)

ax-4bx=3a+b

or a/b=(4x+1)/(x-3)

log6/log3=1.631

1.631x - 4.89=4x+1


-2.369x=5.89


x=-2.49

probably made a few mistakes along the way
but the principle is ok

Answer 4

6^x/6^3 =(3^4)^x*3 so
(6/81)^x = 648 and taking log
x= log648/log(6/81)=-2.4874