Fill in the missing values to make the equations true.
(a) [tex]$\log _2 5+\log _2 11=\log _2 \square$[/tex]
(b) [tex]$\log _5 9-\log _5 \square=\log _5 \frac{9}{11}$[/tex]
(c) [tex]$\log _9 25=\square \log _9 5$[/tex]
Answer (2)
Use the logarithm property lo g a x + lo g a y = lo g a ( x y ) to find the missing value in equation (a): 55 .
Use the logarithm property lo g a x − lo g a y = lo g a ( y x ) to find the missing value in equation (b): 11 .
Use the logarithm property a lo g b x = lo g b x a to find the missing value in equation (c): 2 .
Explanation
Understanding the Problem We are given three equations involving logarithms with missing values that we need to find. We will use the properties of logarithms to simplify the equations and find the missing values.
Solving equation (a) (a) We have the equation lo g 2 5 + lo g 2 11 = lo g 2 □ . Using the property lo g a x + lo g a y = lo g a ( x y ) , we can rewrite the left side as lo g 2 ( 5 × 11 ) = lo g 2 55 . Therefore, the missing value is 55.
Solving equation (b) (b) We have the equation lo g 5 9 − lo g 5 □ = lo g 5 11 9 . Using the property lo g a x − lo g a y = lo g a ( y x ) , we can rewrite the left side as lo g 5 ( □ 9 ) = lo g 5 11 9 . Therefore, □ 9 = 11 9 , which implies that the missing value is 11.
Solving equation (c) (c) We have the equation lo g 9 25 = □ lo g 9 5 . Using the property a lo g b x = lo g b x a , we can rewrite the right side as lo g 9 5 □ . Therefore, lo g 9 25 = lo g 9 5 □ . Since 25 = 5 2 , we have 5 2 = 5 □ , which implies that the missing value is 2.
Final Answer Therefore, the missing values are 55, 11, and 2.
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). Understanding the properties of logarithms allows us to manipulate and solve equations in these contexts. For example, if we know the intensities of two earthquakes, we can use logarithms to compare their magnitudes. Similarly, in finance, logarithms are used to calculate compound interest and analyze investment growth.
The missing values are 55 for equation (a), 11 for equation (b), and 2 for equation (c. These solutions are derived using properties of logarithms to combine and manipulate them accurately. By applying the appropriate logarithmic identities, we can find the unknowns in each equation.
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