Fill in the missing values to make the equations true.
(a) [tex]$\log _2 5+\log _2 11=\log _2$[/tex] ?
(b) [tex]$\log _5 9-\log _5$[/tex] ? [tex]$=\log _5 \frac{9}{11}$[/tex]
(c) [tex]$\log _9 25=$[/tex] ? [tex]$\log _9 5$[/tex]
Answer (1)
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): lo g 2 5 + lo g 2 11 = lo g 2 ( 5 ⋅ 11 ) = lo g 2 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): lo g 5 9 − lo g 5 □ = lo g 5 11 9 implies lo g 5 □ 9 = lo g 5 11 9 , so □ = 11 .
Use the logarithm property x lo g a y = lo g a y x to find the missing value in equation (c): lo g 9 25 = □ lo g 9 5 implies lo g 9 5 2 = □ lo g 9 5 , so 2 lo g 9 5 = □ lo g 9 5 , thus □ = 2 .
The missing values are 55 , 11 , 2 .
Explanation
Understanding the Problem We are given three equations with missing values that we need to find. These equations involve logarithms, and we will use properties of logarithms to find the missing values.
Solving equation (a) (a) We have the equation lo g 2 5 + lo g 2 11 = lo g 2 部. Using the logarithm 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 logarithm 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, the missing value is 11.
Solving equation (c) (c) We have the equation lo g 9 25 = □ lo g 9 5 . We can rewrite 25 as 5 2 , so the equation becomes lo g 9 5 2 = □ lo g 9 5 . Using the logarithm property x lo g a y = lo g a y x , we can rewrite the left side as 2 lo g 9 5 = □ lo g 9 5 . Therefore, the missing value is 2.
Final Answer The missing values are (a) 55, (b) 11, and (c) 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 the time it takes for an investment to double at a given interest rate.
