Question

I need help with number 5

Answer 1

k = 2

Explanation:

To find a value of k such that the equation log_k(8x - 2x²) = 3 has a single solution, we can use the property of logarithms that states:

`\large\boxed{\log_a(b) = c \) \iff \( a^c = b}`

So, for log_k(8x - 2x²) = 3, we have:

`k^3 = 8x - 2x^2`

Rearrange the equation so that is in the form ax² + bx + c = 0:

`2x^2-8x+k^3=0`

In this case:

• a = 2
• b = -8
• c = k³

For a quadratic equation to have a single solution, the discriminant (b² - 4ac) must be equal to zero. Therefore:

`(-8)^2-4(2)(k^3)=0`

Solve for k:

`64-8k^3=0`

`8(8-k^3)=0`

`8-k^3=0`

`k^3=8`

`k^3=2^3`

`k=2`

Therefore, the value of k is 2.