Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Evaluate logarithmic expressions if possible.
[tex]8 \log _b x+5 \log _b z[/tex]
Answer (2)
Apply the power rule of logarithms: 8 lo g b x = lo g b x 8 and 5 lo g b z = lo g b z 5 .
Substitute these back into the original expression: lo g b x 8 + lo g b z 5 .
Apply the product rule of logarithms: lo g b x 8 + lo g b z 5 = lo g b ( x 8 z 5 ) .
The final condensed expression is lo g b ( x 8 z 5 ) .
Explanation
Understanding the problem We are given the expression 8 lo g b x + 5 lo g b z and we want to condense it into a single logarithm with a coefficient of 1. We will use the properties of logarithms to achieve this.
Applying the power rule First, we use the power rule of logarithms, which states that a lo g b x = lo g b x a . Applying this rule to both terms in the expression, we get:
8 lo g b x = lo g b x 8
and
5 lo g b z = lo g b z 5
Substituting back into the expression Now, we substitute these back into the original expression:
lo g b x 8 + lo g b z 5
Applying the product rule Next, we use the product rule of logarithms, which states that lo g b x + lo g b y = lo g b ( x y ) . Applying this rule, we get:
lo g b x 8 + lo g b z 5 = lo g b ( x 8 z 5 )
Final Answer Therefore, the condensed expression is lo g b ( x 8 z 5 ) .
Examples
Logarithms are used in many scientific and engineering fields. For example, they are used to measure the intensity of earthquakes (Richter scale) and the loudness of sounds (decibels). The properties of logarithms, like the ones used in this problem, are essential for simplifying calculations and understanding these scales. In computer science, logarithms are used to analyze the efficiency of algorithms. Condensing logarithmic expressions can help simplify complex formulas and make them easier to work with.
By using the power and product rules of logarithms, the expression 8 lo g b x + 5 lo g b z condenses to lo g b ( x 8 z 5 ) . This means that instead of having two logarithmic terms, we can represent them as a single logarithm. This simplification is useful for working with logarithmic equations more easily.
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