Find all values of c such that
3^(2c+1)=28*(3^c)-9. If you find more than one value of c, then list your values in increaing order, separated by commas.
To find all values of c that satisfy the given equation, we'll need to solve it step-by-step.
Step 1: Start with the given equation.
3^(2c+1) = 28 * (3^c) - 9
Step 2: Simplify the equation.
3^(2c+1) = 28 * 3^c - 9
Step 3: Use the properties of exponents to combine like terms.
3^(2c+1) = 28 * 3^c - 9
3^(2c+1) = 3^1 * 3^c * 28 - 9
3^(2c+1) = 3^(c+1) * 28 - 9
Step 4: Since the bases are the same (3), we can set the exponents equal to each other.
2c + 1 = c + 1
Step 5: Solve for c.
2c - c = 1 - 1
c = 0
Step 6: Check if the solution is valid.
Substitute c = 0 back into the original equation:
3^(2c+1) = 28 * 3^c - 9
3^(2*0+1) = 28 * 3^0 - 9
3^1 = 28 * 1 - 9
3 = 28 - 9
3 = 19
The equation is not satisfied when c = 0. Therefore, there are no values of c that satisfy the given equation.
To solve the equation 3^(2c+1) = 28 * (3^c) - 9 and find the values of c, you can follow these steps:
Step 1: Simplify the equation
Using the properties of exponential equations, we can rewrite the equation as follows:
3^(2c) * 3^1 = 28 * 3^c - 9
Step 2: Expand the equation
Using the properties of exponentials, we can rewrite the equation further:
9 * 3^c = 28 * 3^c - 9
Step 3: Move the terms involving 3^c to one side
Subtract 28 * 3^c from both sides of the equation:
9 * 3^c - 28 * 3^c = -9
Step 4: Combine like terms
Factor out 3^c from the left side of the equation:
(9 - 28) * 3^c = -9
-19 * 3^c = -9
Step 5: Solve for 3^c
Divide both sides of the equation by -19:
3^c = -9 / -19
3^c = 9/19
Step 6: Solve for c using logarithms
To find the value of c, we can take the logarithm of both sides of the equation:
log base 3 of (3^c) = log base 3 of (9/19)
Using the logarithmic property log base b of (a^c) = c * log base b of a:
c * log base 3 of 3 = log base 3 of (9/19)
c = log base 3 of (9/19)
Using logarithm properties, we can evaluate the logarithm expression to find the value of c.
Hence, the value of c is log base 3 of (9/19).
3^(2c+1)=28*(3^c)-9
3^(2c)*3^1 = 28*3^c - 9
3(3^c)^2 - 28(3^c) + 9 = 0
let 3^c = x
3x^2 - 28x + 9 = 0
(3x - 1)(x - 9) = 0
x = 1/3 or x = 9
if x = 1/3
3^c= 1/3 = 3^-1 -----> c = -1
if x = 9
3^c = 9 = 3^2 ----> c = 2