Use the properties of logarithms to condense the following expression into a single logarithm. Be sure to leave the numerator and denominator in factored form.
[tex]2 \log _8(3 x)-\log _8(x+4)+\log _8(x-5)[/tex]
Answer (1)
Apply the power rule: 2 lo g 8 ( 3 x ) = lo g 8 ( 9 x 2 ) .
Apply the quotient rule: lo g 8 ( 9 x 2 ) − lo g 8 ( x + 4 ) = lo g 8 ( x + 4 9 x 2 ) .
Apply the product rule: lo g 8 ( x + 4 9 x 2 ) + lo g 8 ( x − 5 ) = lo g 8 ( x + 4 9 x 2 ( x − 5 ) ) .
The condensed expression is lo g 8 ( x + 4 9 x 2 ( x − 5 ) ) .
Explanation
Problem Analysis We are asked to condense the expression 2 lo g 8 ( 3 x ) − lo g 8 ( x + 4 ) + lo g 8 ( x − 5 ) into a single logarithm, ensuring the numerator and denominator are in factored form. We will use the properties of logarithms to achieve this.
Applying the Power Rule First, we apply the power rule of logarithms, which states that a lo g b ( x ) = lo g b ( x a ) . Applying this to the first term, we get: 2 lo g 8 ( 3 x ) = lo g 8 (( 3 x ) 2 ) = lo g 8 ( 9 x 2 ) So the expression becomes: lo g 8 ( 9 x 2 ) − lo g 8 ( x + 4 ) + lo g 8 ( x − 5 )
Applying the Quotient Rule Next, we use the quotient rule of logarithms, which states that lo g b ( x ) − lo g b ( y ) = lo g b ( y x ) . Applying this to the first two terms, we have: lo g 8 ( 9 x 2 ) − lo g 8 ( x + 4 ) = lo g 8 ( x + 4 9 x 2 ) Now the expression is: lo g 8 ( x + 4 9 x 2 ) + lo g 8 ( x − 5 )
Applying the Product Rule Now, 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 to the remaining terms, we get: lo g 8 ( x + 4 9 x 2 ) + lo g 8 ( x − 5 ) = lo g 8 ( x + 4 9 x 2 ( x − 5 ) ) Thus, the condensed expression is: lo g 8 ( x + 4 9 x 2 ( x − 5 ) )
Final Answer The expression is now condensed into a single logarithm. The numerator 9 x 2 ( x − 5 ) and the denominator x + 4 are both in factored form. Therefore, the final answer is: lo g 8 ( x + 4 9 x 2 ( x − 5 ) )
Examples
Logarithms are used in many scientific and engineering fields. For example, in acoustics, the loudness of a sound is measured in decibels using a logarithmic scale. Compressing the expression helps in simplifying calculations involving sound intensities. Similarly, in finance, logarithmic scales are used to analyze investment growth and risk. By condensing logarithmic expressions, financial analysts can more easily compare different investment options and make informed decisions.
