Final answer:
To solve the equation 6^(2x+1) = 2^(3x+1), first rewrite 6 as 2^1x3 and simplify. Then, equate the exponents on both sides and solve for x. So, either x = 0 or 2log3 + 1 = 0.
Step-by-step explanation:
To solve the equation 6^(2x+1) = 2^(3x+1), we need to equate the exponents on both sides. To do this, we can rewrite 6 as 2^1x3 and simplify the equation:
(2^1x3)^(2x+1) = 2^(3x+1)
Using the property of exponents that says (a^m)^n = a^(mn), we can rewrite the equation as:
2^(2x+1)x3^(2x+1) = 2^(3x+1)
Now, we can equate the exponents:
2x + 1 + (2x+1)log3 = 3x + 1
Simplifying, we get:
4x + 1 + 2xlog3 = 3x + 1
Subtracting 1 from both sides:
4x + 2xlog3 = 3x
Subtracting 3x from both sides:
x(4 + 2log3 - 3) = 0
Simplifying further:
x(2log3 + 1) = 0
So, either x = 0 or 2log3 + 1 = 0.