Simplify. Enter the result as a single logarithm with a coefficient of 1.
[tex]$\log _9(8 x^5)-\log _9(7 x^6)=$\n$\square$[/tex]
Answer (2)
Use the logarithm property lo g b ( A ) − lo g b ( B ) = lo g b ( B A ) to combine the logarithms.
Simplify the fraction inside the logarithm.
The simplified expression is lo g 9 ( 7 x 8 ) .
The final answer is lo g 9 ( 7 x 8 ) .
Explanation
Understanding the Problem We are given the expression lo g 9 ( 8 x 5 ) − lo g 9 ( 7 x 6 ) . Our goal is to simplify this expression into a single logarithm with a coefficient of 1.
Applying Logarithm Properties We will use the logarithm property that states the difference of two logarithms with the same base is equal to the logarithm of the quotient of their arguments: lo g b ( A ) − lo g b ( B ) = lo g b ( B A )
Combining the Logarithms Applying this property to our expression, we get: lo g 9 ( 8 x 5 ) − lo g 9 ( 7 x 6 ) = lo g 9 ( 7 x 6 8 x 5 )
Simplifying the Fraction Now, we simplify the fraction inside the logarithm: 7 x 6 8 x 5 = 7 8 ⋅ x 6 x 5 = 7 8 ⋅ x 5 − 6 = 7 8 ⋅ x − 1 = 7 x 8
Final Result Therefore, the simplified expression is: lo g 9 ( 7 x 6 8 x 5 ) = lo g 9 ( 7 x 8 )
Examples
Logarithms are used in many scientific and engineering applications, such as measuring the intensity of earthquakes (the Richter scale) or the acidity of a solution (pH scale). Simplifying logarithmic expressions can help in calculations involving these scales. For instance, if you are comparing the intensity of two earthquakes and have their Richter scale values, subtracting the logarithms can give you the ratio of their intensities. This is a direct application of the logarithm property used in this problem.
We simplified the expression lo g 9 ( 8 x 5 ) − lo g 9 ( 7 x 6 ) by using the property of logarithms to combine them as a single logarithm lo g 9 ( 7 x 8 ) . The final result is lo g 9 ( 7 x 8 ) .
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