Exponents and Logarithms, Simplifying and simplest form

Question

Write each expression as a single logarithm in simplest form

Log₅ (x²-16) - Log₅ (x²-2x-8)

3Log₈ (x+3) - Log₈ (x²+x-6)

They have to be in the format that is log_x[x/x]

A simplified fraction in simplest form of those two questions

Answer 1

By the quotient rule:

log₅ (x²-16) - log₅ (x²-2x-8) = log₅ [(x²-16)/(x²-2x-8)]

Now,

x²-16 is a subtraction of squares, so it can be factorized like this:

x²-16 = (x+4)(x-4)

x²-2x-8 can also be factorized:

x²-2x-8 = (x-4)(x+2)

We can simplify the fraction:

(x²-16)/(x²-2x-8)= (x+4)/(x+2)

log₅ [(x²-16)/(x²-2x-8)] = log₅ [(x+4)/(x+2)] is the simplified form of the 1st formula.

However, we need to carry forward the constraints.

The value inside a log must be greater than 0 so:

x²-16 = (x+4)(x-4) > 0

The zeros are at: -4, 4

If we sample a value at x = 0, we get a negative value. So, the function must be positive at:

x < -4, x > 4

and

x²-2x-8 = (x-4)(x+2) > 0

The zeros are at: -2, 4

If we sample a value at x = 0, we get a negative value, So, the function must be positive at:

x < -2, x > 4

The overall constraint must satisfy both constraints, so: x < -4, x > 4

Note that the zeros of the denominator are already handled by this constraint because x cannot equal 4 or -2.