Question

Which of these expressions is equivalent to log (46)? A. log (6) log (4) B. log (6) - log (4) C. 6. log (4) D. log (6) + log (4)

Answer 1

Option D. log (6) + log (4) because of the logarithmic property known as the product rule.

Step-by-step explanation:

Logarithmic properties allow for the manipulation of logarithms through addition, subtraction, and multiplication. The expression log(46) can be broken down using the product rule of logarithms, which states that log(a * b) = log(a) + log(b). Applying this rule, log(46) can be rewritten as log(6 * 4).

This translates to log(6) + log(4), hence Option D is equivalent to log(46). In detail, logarithms express exponents, so log(46) can be seen as log(2 * 23), which can further be written as log(2) + log(23).

However, 23 is 2 * 11.5, but log(11.5) isn't a straightforward integer or commonly simplified fraction. Instead, it's easier to express 23 as 2 * 11.5 and represent log(46) as log(2) + log(2 * 11.5), which simplifies to log(2) + (log(2) + log(11.5)). Further, log(2) + log(2) equals 2 * log(2), hence log(46) equals log(2) + 2 * log(2). This reduces to log(6) + log(4).

Therefore, by applying logarithmic properties and simplifying the expression step by step, it's evident that log(46) is equivalent to log(6) + log(4), aligning with Option D.

Answer 2

option D:
`\( \log(6) + \log(4) \)`.

The logarithmic identity that is relevant here is:

`\[ \log_a(b) + \log_a(c) = \log_a(bc) \]`

Therefore, we can use this property to rewrite the given expression:

`\[ \log(46) = \log(6 \cdot 4) \]`

Now, let's compare the given options:

A.
`\( \log(6) \cdot \log(4) \)` - Not equivalent.

B.
`\( \log(6) - \log(4) \)` - Not equivalent.

C.
`\( 6 \cdot \log(4) \)` - Not equivalent.

D.
`\( \log(6) + \log(4) \)` - Equivalent.