analyze the function ln x = cx^2 to find the unique value of c such that there is exactly one solution to the equation. to do this, find the value of c such that both sides of the equation have equivalent slopes at some point; this will give you a proper x-coordinate to work with.

To find the unique value of c such that there is exactly one solution to the equation ln(x) = cx^2, we will analyze the slopes of both sides of the equation.

The slope of the function ln(x) can be found by differentiating the function with respect to x. The derivative of ln(x) is 1/x.

The slope of the function cx^2 can be found by differentiating with respect to x. The derivative of cx^2 is 2cx.

For the two functions to have equivalent slopes at some point, the derivatives must be equal.

So, we set the derivatives equal to each other: 1/x = 2cx.

Now we can solve for x to find the proper x-coordinate. Rearrange the equation to isolate x:

1 = 2cx^2

Divide both sides by 2c:

1/(2c) = x^2

Take the square root of both sides:

sqrt(1/(2c)) = x

Now we have the x-coordinate we need to compare the slopes of the two functions.

Substitute this x-coordinate back into the derivative of ln(x) to find the slope:

1/x = 1/sqrt(1/(2c))

Simplify the expression by multiplying the numerator and denominator by sqrt(2c):

1/x = sqrt(2c) / sqrt(1)

1/x = sqrt(2c)

Now we have the slope of the ln(x) function at the x-coordinate we found.

Substitute the x-coordinate into the derivative of cx^2 to find the slope:

2cx = 2c * sqrt(1/(2c))

Simplify the expression by multiplying the numerator and denominator by sqrt(2c):

2cx = 2c * sqrt(2c) / sqrt(1)

2cx = 2c * sqrt(2c)

Now we have the slope of the cx^2 function at the x-coordinate we found.

For both slopes to be equal, we set the two expressions equal to each other:

sqrt(2c) = 2c * sqrt(2c)

To analyze this equation, we can square both sides to remove the square root:

( sqrt(2c) )^2 = (2c * sqrt(2c) )^2

2c = (2c * sqrt(2c) )^2

Simplify the equation:

2c = 4c^2 * 2c

Divide both sides by 2c:

1 = 4c^2

Rearrange the equation:

4c^2 - 1 = 0

This is a quadratic equation. We can solve it by factoring:

(2c + 1)(2c - 1) = 0

The two possible solutions for c are c = -1/2 and c = 1/2.

However, we were looking for the unique value of c such that there is exactly one solution to the equation. To find this, we need to check whether the two slopes are equal at the x-coordinate for both values of c.

Substitute the x-coordinate we found earlier (sqrt(1/(2c))) into both derivatives:

1 / sqrt(1/(2c)) = sqrt(2c) (for c = -1/2)
1 / sqrt(1/(2c)) = sqrt(2c) (for c = 1/2)

In both cases, we obtain the same equation, which means that both slopes are equal at that x-coordinate.

Therefore, the unique value of c such that there is exactly one solution to the equation ln(x) = cx^2 is c = -1/2.