Completely expand the logarithm: $n \log _b \frac{x}{y z}$
Answer (2)
Apply the quotient rule: n l o g b f r a c x yz = n ( l o g b x − l o g b ( yz )) .
Apply the product rule: l o g b ( yz ) = l o g b y + l o g b z .
Distribute the negative sign: n ( l o g b x − l o g b y − l o g b z ) .
Distribute n : n l o g b x − n l o g b y − n l o g b z . The final answer is n l o g b x − n l o g b y − n l o g b z .
Explanation
Understanding the Problem We are given the expression n l o g b f r a c x yz and asked to expand it completely. To do this, we will use the properties of logarithms.
Applying the Quotient Rule First, we use the quotient rule of logarithms, which states that l o g b f r a c A B = l o g b A − l o g b B . Applying this rule to our expression, we get:
n l o g b f r a c x yz = n ( l o g b x − l o g b ( yz ))
Applying the Product Rule Next, we use the product rule of logarithms, which states that l o g b ( A B ) = l o g b A + l o g b B . Applying this rule to the term l o g b ( yz ) , we get:
l o g b ( yz ) = l o g b y + l o g b z
Substituting Back Now, we substitute this back into our expression:
n ( l o g b x − ( l o g b y + l o g b z ))
Distributing the Negative Sign Distribute the negative sign:
n ( l o g b x − l o g b y − l o g b z )
Distributing n and Concluding Finally, distribute the n to each term:
n l o g b x − n l o g b y − n l o g b z
Therefore, the completely expanded form of the given logarithm is n l o g b x − n l o g b y − n l o g b z .
Examples
Logarithms are used in many scientific and engineering fields. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. The formula is M = l o g 10 A − l o g 10 A 0 , where M is the magnitude, A is the amplitude of the seismic waves, and A 0 is a reference amplitude. Expanding logarithmic expressions helps in simplifying and analyzing such formulas.
To expand the logarithm n lo g b yz x , we use the quotient rule to separate it into n ( lo g b x − lo g b ( yz )) and then apply the product rule to express lo g b ( yz ) as lo g b y + lo g b z . The final result is n lo g b x − n lo g b y − n lo g b z .
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