Condense to a single logarithm: [tex]ln x+(1 / 3) ln y+(1 / 3) ln z[/tex].
Answer (2)
Apply the power rule of logarithms to rewrite the terms: ( 1/3 ) ln y = ln y 1/3 and ( 1/3 ) ln z = ln z 1/3 .
Rewrite the expression as ln x + ln y 1/3 + ln z 1/3 .
Apply the product rule of logarithms to combine the terms: ln x + ln y 1/3 + ln z 1/3 = ln ( x y 1/3 z 1/3 ) .
The condensed expression is ln ( x y 1/3 z 1/3 ) .
Explanation
Understanding the Problem We are asked to condense the expression ln x + ( 1/3 ) ln y + ( 1/3 ) ln z into a single logarithm. We will use properties of logarithms to combine the terms.
Applying the Power Rule First, we use the power rule of logarithms, which states that a ln b = ln b a . Applying this rule to the second and third terms, we get: ( 1/3 ) ln y = ln y 1/3 ( 1/3 ) ln z = ln z 1/3
Rewriting the Expression Now we rewrite the expression as: ln x + ln y 1/3 + ln z 1/3
Applying the Product Rule Next, we use the product rule of logarithms, which states that ln a + ln b = ln ( ab ) . Applying this rule to combine the terms, we have: ln x + ln y 1/3 + ln z 1/3 = ln ( x y 1/3 z 1/3 )
Final Answer Therefore, the condensed expression is ln ( x y 1/3 z 1/3 ) .
Examples
Logarithms are used in many scientific and engineering applications, such as calculating the magnitude of earthquakes (Richter scale), measuring sound intensity (decibels), and modeling population growth. Condensing logarithmic expressions can simplify complex calculations and make them easier to interpret. For example, in acoustics, combining multiple sound sources into a single logarithmic expression can help determine the overall sound level.
By utilizing the properties of logarithms, we can condense the expression ln x + 3 1 ln y + 3 1 ln z into a single logarithm, resulting in ln ( x y 1/3 z 1/3 ) .
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