Express as a single logarithm and, if possible, simplify:
[tex]$\frac{1}{2} \log b+3 \log c$[/tex]
[tex]$\frac{1}{2} \log b+3 \log c=\square$[/tex]
(Use parentheses to indicate the argument of the logarithm.)
Answer (2)
Apply the power rule of logarithms: 2 1 lo g b = lo g b 2 1 and 3 lo g c = lo g c 3 .
Substitute back into the original expression: lo g b 2 1 + lo g c 3 .
Apply the product rule of logarithms: lo g ( b 2 1 c 3 ) .
Rewrite b 2 1 as b : lo g ( b c 3 ) . The final answer is lo g ( b c 3 ) .
Explanation
Understanding the problem We are given the expression 2 1 lo g b + 3 lo g c and we want to express it as a single logarithm. We will use logarithm properties to achieve this.
Applying the power rule First, we use the power rule of logarithms, which states that a lo g x = lo g x a . Applying this rule, we have: 2 1 lo g b = lo g b 2 1 3 lo g c = lo g c 3
Substituting back Now, we substitute these back into the original expression: lo g b 2 1 + lo g c 3
Applying the product rule Next, we use the product rule of logarithms, which states that lo g x + lo g y = lo g ( x y ) . Applying this rule, we have: lo g b 2 1 + lo g c 3 = lo g ( b 2 1 c 3 ) We can rewrite b 2 1 as b . Therefore, lo g ( b 2 1 c 3 ) = lo g ( b c 3 )
Final Answer Thus, the expression 2 1 lo g b + 3 lo g c can be expressed as a single logarithm as lo g ( b c 3 ) .
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 sound (decibels). The properties of logarithms, such as the power rule and product rule, are useful in simplifying complex expressions and solving equations in these fields. Understanding how to combine and simplify logarithmic expressions can help in analyzing and interpreting data in various real-world applications.
The expression 2 1 lo g b + 3 lo g c can be expressed as a single logarithm using logarithm properties. By applying the power rule and product rule, we arrive at the simplified form lo g ( b c 3 ) . The final answer is lo g ( b c 3 ) .
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