Expand the following logarithmic expression.

$\log _5\left(8 \frac{\sqrt{t}}{v}\right)$

Apply the product rule: lo g 5 ​ ( 8 v t ​ ​ ) = lo g 5 ​ ( 8 ) + lo g 5 ​ ( v t ​ ​ ) .
Apply the quotient rule: lo g 5 ​ ( 8 ) + lo g 5 ​ ( t ​ ) − lo g 5 ​ ( v ) .
Rewrite the square root as a power: lo g 5 ​ ( 8 ) + lo g 5 ​ ( t 1/2 ) − lo g 5 ​ ( v ) .
Apply the power rule: lo g 5 ​ ( 8 ) + 2 1 ​ lo g 5 ​ ( t ) − lo g 5 ​ ( v ) . The final answer is lo g 5 ​ 8 + 2 1 ​ lo g 5 ​ t − lo g 5 ​ v ​ .

Explanation

Understanding the Problem We are given the logarithmic expression lo g 5 ​ ( 8 v t ​ ​ ) . Our goal is to expand this expression using the properties of logarithms.

Logarithm Properties We will use the following logarithm properties:

Product Rule: lo g b ​ ( x y ) = lo g b ​ ( x ) + lo g b ​ ( y )

Quotient Rule: lo g b ​ ( y x ​ ) = lo g b ​ ( x ) − lo g b ​ ( y )

Power Rule: lo g b ​ ( x n ) = n lo g b ​ ( x )

Expanding the Expression First, we apply the product rule to the expression: lo g 5 ​ ( 8 v t ​ ​ ) = lo g 5 ​ ( 8 ) + lo g 5 ​ ( v t ​ ​ ) Next, we apply the quotient rule to the second term: lo g 5 ​ ( 8 ) + lo g 5 ​ ( t ​ ) − lo g 5 ​ ( v ) Now, we rewrite the square root as a power: lo g 5 ​ ( 8 ) + lo g 5 ​ ( t 1/2 ) − lo g 5 ​ ( v ) Finally, we apply the power rule to the second term: lo g 5 ​ ( 8 ) + 2 1 ​ lo g 5 ​ ( t ) − lo g 5 ​ ( v )

Final Answer Comparing our expanded expression with the given options, we find that it matches the last option: lo g 5 ​ 8 + 2 1 ​ lo g 5 ​ t − lo g 5 ​ v


Examples
Logarithmic expressions are used in many fields, including physics, engineering, and computer science. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. Expanding logarithmic expressions can help simplify calculations and make it easier to understand the relationships between different variables. In finance, logarithmic transformations are often used to analyze investment returns and model asset prices. Understanding how to manipulate logarithmic expressions is a valuable skill in many areas of study and work.

Answered by GinnyAnswer | 2025-07-08