Question

Express as a sum of logarithms
Log4(176*2)

Answer 1

`log_(4)(176*2)=log_(4)176 +log_(4)2`

Explanation:

`log_(x)(A*B)=log_(x)A +log_(x)B`

Applying this logarithms property to the given example, we get:

`log_(4)(176*2)=log_(4)176 +log_(4)2`

Answer 2

log₄(176) +log₄(2) = 2.5 +log₄(11)

Explanation:

You want log₄(176·2) expressed as the sum of logarithms.

Rules of logarithms

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

log(a^b) = b·log(a)

log₄(4) = 1

Application

Using the log of the product, we have ...

log₄(176·2) = log₄(176) +log₄(2)

This can be further simplified by factoring out powers of 2 (or 4).

log₄(176) +log₄(2) = log₄(4²·11) +log₄(4^(1/2))

= 2 +log₄(11) +1/2

= 2.5 +log₄(11)

__

Additional comment

We have assumed the conventional meaning of the asterisk (*) in your expression. If you intend it as an exponent indicator, then the result is different:

log₄(176²) = 2·log₄(176) = 2(2 +log₄(11))

= 4 +2·log₄(11)