Question

Help fast, ​please

A. Expand the following and state the Law that is indicated.

1. log4(3x)

2. log3(27/x)

3. log4(x5)

Answer 1

1.

`log_(4)(3x) = log_(4)(3) + log_(4)(x)`

2.

`log_(3)( (27)/(x) ) = 3 - log_(3)(x)`

3.

`log_(4)( {x}^(5) ) = 5 log_(4)(x)`

Step-by-step explanation

1. The given logarithmic expression is

`log_(4)(3x)`

Use the product rule:

`log_(a)(mn) = log_(a)(m) + log_(a)(n)`

We apply this rule to obtain:

`log_(4)(3x) = log_(4)(3) + log_(4)(x)`

2. The given logarithmic expression is

`log_(3)( (27)/(x) )`

We apply the quotient rule:

`log_(a)( (m)/(n) ) = log_(a)(m) - log_(a)(n)`

This implies that;

`log_(3)( (27)/(x) ) = log_(3)(27) - log_(3)(x)`

We simplify to get;

`log_(3)( (27)/(x) ) = log_(3)( {3}^(3) ) - log_(3)(x)`

Apply the power rule:

`log_(a)( {m}^(n) ) = n log_(a)(m)`

`log_(3)( (27)/(x) ) = 3 log_(3)( {3}) - log_(3)(x)`

simplify;

`log_(3)( (27)/(x) ) = 3 (1) - log_(3)(x)`

`log_(3)( (27)/(x) ) = 3 - log_(3)(x)`

3. The given logarithmic expression is;

`log_(4)( {x}^(5) )`

Apply the power rule of logarithms.

`log_(a)( {m}^(n) ) = n log_(a)(m)`

This implies that,

`log_(4)( {x}^(5) ) = 5 log_(4)(x) .`