To determine

To evaluate: The value of  b

Answer

b=e2

Explanation

Given that:

1. f(x)=logb(3x2−2)

2. f'(1)=3

Formula used:

1. Chain rule: dfdx=dfdududx

2. ddx(logbx)=1xlnb

Calculation:

For finding value of b, we have to use condition f'(1)=3. So, we have to first find out the value of the derivative of f. Here, f  is composition of functions, so to differentiate f  we have to apply chain rule. For applying the chain rule, let u=3x2−2. Then, f(x)=logbu.

Differentiate it with respect to x, we get

f'(x)=ddx(logbu)=ddu(logbu)dudx                                                   (By using chain rule)=1ulnbddx(u)                                                        (By formula 2)=1lnb(3x2−2)ddx(3x2−2)                                   (By replacing value u=3x2−2)=1lnb(3x2−2)(6x−0)                                   =6xlnb(3x2−2)

Put x=1 in above, we get

f'(1)=6lnb(3−2)         =6lnb

But, it is given that f'(1)=3. Therefore,

3=6lnbThis implies that lnb=2  which is equivalent to b=e2

Conclusion: Thus, b=e2.