7 raise to power 2x -2(7 raise to power x) +1=0.
To solve the equation 7^(2x) - 2(7^x) + 1 = 0, let's make a substitution:
Let u = 7^x.
Substituting this into the equation, we get u^2 - 2u + 1 = 0.
Now we can solve this quadratic equation:
(u - 1)(u - 1) = 0
(u - 1)^2 = 0
Taking the square root of both sides, we have:
u - 1 = 0
u = 1
Substituting back u = 7^x, we have:
7^x = 1
Since any number raised to the power of 0 is equal to 1, we have:
x = 0
To solve the equation 7^(2x) - 2*7^x + 1 = 0, we can use a substitution.
Let's substitute a variable to simplify the equation. Let's set u = 7^x.
Now, rewrite the equation with this substitution:
u^2 - 2u + 1 = 0.
This is now a quadratic equation that can be factored:
(u-1)(u-1) = 0.
Since (u-1)^2=0, it means that u-1=0, which implies u=1.
Now substitute back the value of u:
7^x = 1.
To solve for x, we need to take the logarithm of both sides of the equation.
Take the logarithm base 7 of both sides:
log7(7^x) = log7(1).
The log7(7^x) simplifies to x, and log7(1) is equal to 0. So, the equation becomes:
x = 0.
Therefore, the solution to the equation 7^(2x) - 2*7^x + 1 = 0 is x = 0.
To solve the equation 7^(2x) - 2(7^x) + 1 = 0, we can make a substitution to simplify the equation. Let's say we substitute a variable, let's say y, for 7^x.
So, let y = 7^x.
Now we can rewrite the equation in terms of y:
y^2 - 2y + 1 = 0.
This is a quadratic equation. Let's solve it using factoring:
(y - 1)(y - 1) = 0.
Now set each factor equal to zero:
y - 1 = 0 or y - 1 = 0.
Solve for y:
y = 1 or y = 1.
Now, substitute back in the original variable:
7^x = 1 or 7^x = 1.
Since any number raised to the power of zero is equal to 1, we can write the solutions as:
7^x = 1.
Now, there are two cases to consider:
Case 1: 7^x = 1
In this case, x would be equal to 0 because any number raised to the power of 0 is 1.
Case 2: 7^x = 1
In this case, x would be equal to log base 7 of 1. Since any number raised to the power of zero is 1, we can write this as:
x = log 1 / log 7
The logarithm of 1 to any base is always 0, so we get:
x = 0 / log 7
Therefore, the two solutions to the given equation are:
x = 0 or x = 0 / log 7.