Condense the expression.
[tex]$\begin{array}{c}
\log 5-2 \log x \\
\log [?]
\end{array}$[/tex]
Answer (1)
Use the power rule to rewrite 2 lo g x as lo g x 2 .
Apply the quotient rule to combine the logarithms: lo g 5 − lo g x 2 = lo g x 2 5 .
The condensed expression is lo g x 2 5 .
Explanation
Understanding the Problem We are asked to condense the expression lo g 5 − 2 lo g x into a single logarithm. This involves using properties of logarithms to combine the terms.
Applying the Power Rule First, we use the power rule of logarithms, which states that a lo g b = lo g b a . Applying this rule to the second term, we have 2 lo g x = lo g x 2 .
Applying the Quotient Rule Now, we have lo g 5 − lo g x 2 . We can use the quotient rule of logarithms, which states that lo g a − lo g b = lo g b a . Applying this rule, we get lo g 5 − lo g x 2 = lo g x 2 5 .
Final Answer Therefore, the condensed expression is lo g x 2 5 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (Richter scale), the loudness of sound (decibels), and the acidity of a solution (pH scale). Condensing logarithmic expressions can simplify calculations in these applications. For example, if you are comparing the intensities of two earthquakes, you might need to simplify a logarithmic expression to find the ratio of their intensities. Understanding how to manipulate logarithms is essential in these fields.
