Question

Maths log question please help- image attached

Answer 1

Answer:

a = 2 and b = -1.

Step-by-step explanation:

We want to show that the equation:

$\displaystyle \log_{2} \left( y + 1\right) - 1 = 2\log_{2} x$

Can be rewritten as:

$\displaystyle y = ax^2 + b$

Where a and b are integers.

Note that:

$\displaystyle 1 = \log _{2} 2$

Hence:

$\displaystyle \log_{2} \left( y + 1\right) - \log _2 2= 2\log_{2} x$

From logarithmic properties:

$\displaystyle \log_2 \left(\frac{y+1}{2}\right) = 2\log_2 x$

Likewise:

$\displaystyle \log_2 \left(\frac{y+1}{2}\right) = \log_2 x^2$

For the logs to be equivalent, the inner equations must be equivalent. Hence:

$\displaystyle \frac{y+1}{2} = x^2$

Simplify:

$\displaystyle y = 2x^2 - 1$

In conclusion, a = 2 and b = -1.