Precalculus Question

Question

The expression ion log (x^10 y^16 / z^16) can be written in the form,

A log(x) + B log(y) + C log(z)

Where

A = ___?

B = ___?

C = ___?

Answer 1

Logarithms are fanciful ways to reduce the operation type to allow for manipulating large and small numbers more efficiently.

log₁₀(100) = log₁₀(10²) = 2

They reduce exponentiation to multiplication/division and change multiplication/division to addition/subtraction

So,

log(x¹⁰y¹⁶/z¹⁶) can be seen as a multiplication and a division. Multiplication becomes addition and division becomes subtraction...

log(x¹⁰y¹⁶/z¹⁶) = log(x¹⁰) + log(y¹⁶) - log(z¹⁶)

Now exponentiation is reduced (as shown in the example above) by bring down the exponent to become a multiplier.

log(x¹⁰y¹⁶/z¹⁶) = log(x¹⁰) + log(y¹⁶) - log(z¹⁶) = 10log(x) + 16log(y) - 16log(z)

So A = 10, B = 16, and C = -16 when fit to the expression Alog(x) + Blog(y) + Clog(z)