Question

3) find x if log³(x+6)=2​

Answer 1

Answer:

If you can recall, there exists a property based on logarithmic functions, which states that for any base of logarithm of product:

log (ab) = (log a) + (log b)

This question is based on this property, expanding your numerical, we get

log 3 + log (1-x) = log 3 + log (x+16-x²)

So, log (1-x) = log (x+16-x²)

Taking antilog on both sides above,

1-x = x+16-x²

x²-2x-15=0

x²-5x+3x-15=0

x(x-5) + 3(x-5)=0

(x-5)(x+3)=0

Hence either x=5 or x=-3

But if we consider x=5,

Then LHS=log3(1–5)=log3(-4)

And logarithmic function takes only positive values in its domain, hence x=5 should not be considered

Therefore x=-3

LHS=log3(1-(-3))=log3(4)=log12

RHS=log3(-3+16-(-3)²)=log3(13–9)=log3(4)=log12

Hence our conditions are satisfied.

x=-3