solve 27^2x-1=(1/81)^x
27=3^3
81=3^4
so
3^(6x-3) = 3^-4x
6x-3 = -4x
x = 3/10
To solve the equation 27^(2x-1) = (1/81)^x, we'll begin by rewriting both sides with the same base.
First, let's express 27 as a power of 3:
27 = 3^3
Now, let's express (1/81) as a power of 3 using the fact that 81 = 3^4:
81 = 3^4
1/81 = 3^(-4)
Rewriting the equation, we have:
(3^3)^(2x-1) = (3^(-4))^x
Using the property (a^m)^n = a^(m*n), we can simplify:
3^(3*(2x-1)) = 3^(-4x)
Now, since the bases are the same, we can equate the exponents:
3*(2x-1) = -4x
Expanding the left side, we get:
6x - 3 = -4x
Combining like terms by adding 4x to both sides:
6x + 4x - 3 = 0
Simplifying:
10x - 3 = 0
Adding 3 to both sides:
10x = 3
Finally, dividing both sides by 10:
x = 3/10
Therefore, the solution to the equation 27^(2x-1) = (1/81)^x is x = 3/10.
To solve the equation 27^(2x-1) = (1/81)^x, we need to use the property of exponents which states that if a^k = b^k, then a = b.
Let's convert both sides of the equation to have the same base.
First, let's express 27 and 1/81 as powers of the same base, which is 3:
27 = 3^3
1/81 = (1/3^4) = 3^(-4)
Now we can rewrite the equation as:
(3^3)^(2x-1) = (3^(-4))^x
Next, apply the exponent rule which states that (a^b)^c = a^(b*c):
3^(3*(2x-1)) = 3^(-4x)
Now that the bases are the same, we can apply the property of exponents that states if a^k = a^m, then k = m:
3*(2x-1) = -4x
Now we simplify the equation:
6x - 3 = -4x
Add 4x to both sides:
6x - 3 + 4x = -4x + 4x
Combine like terms:
10x - 3 = 0
Add 3 to both sides:
10x - 3 + 3 = 0 + 3
Combine like terms:
10x = 3
Finally, divide both sides by 10:
(10x)/10 = 3/10
Simplify:
x = 3/10
Therefore, the solution to the equation 27^(2x-1) = (1/81)^x is x = 3/10.