Expand the expression. [tex]\(\log_2(3x)\)[/tex]
Answer (2)
The expression lo g 2 ( 3 x ) can be expanded using the property of logarithms as lo g 2 ( 3 ) + lo g 2 ( x ) . This is done by recognizing that the logarithm of a product can be broken down into the sum of logarithms of the individual factors. Thus, you can express the original logarithm in a simpler form.
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To expand the expression lo g 2 ( 3 x ) , you can use the logarithmic property known as the 'Product Rule.' The Product Rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
Mathematically, this is expressed as:
lo g b ( mn ) = lo g b ( m ) + lo g b ( n )
In your expression, 3 x is a product of 3 and x . Therefore, using the Product Rule, we can expand lo g 2 ( 3 x ) as follows:
lo g 2 ( 3 x ) = lo g 2 ( 3 ) + lo g 2 ( x )
This expression shows that we have separated the logarithm of the product into the sum of two logarithms, one for each factor of the original product. This is a common technique in algebra to simplify expressions or solve equations involving logarithms.
