Question

Write −2−log4x+1/2log4(x+1)+log4y as a single logarithm with coefficient 1 . Note: If you are using log you need to type it in and use the subscript button on the keyboard. There is no log burton.

Answer 1

log₄(y√(x+1)/(16x))

Explanation:

You want −2−log₄x+1/2log₄(x+1)+log₄y as a single logarithm.

Rules of logarithms

The relevant rules of logarithms are ...

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

log(a/b) = log(a) -log(b)

log(a^b) = b·log(a)

Application

Writing the expression as a sum of logs, we have ...

log₄(4^(-2)) -log₄(x) +log₄(√(x+1)) +log₄(y)

= log₄(y√(x+1)/(16x))

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Additional comment

The only thing under the radical is (x+1).

The logarithm of the base is 1, so ...

log₄(4) = 1

-2 = -2·1 = -2·log₄(4) = log₄(4^-2)

This lets us use base 4 logarithms for everything.

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