Product property:
[tex]$\log _b x y=\log _b x+\log _b y$[/tex]
How would you expand [tex]$\log _4 12$[/tex] so that it can be evaluated, given [tex]$\log _4 3 \approx 0.792$[/tex]?
[tex]$\log _4 3 \cdot \log _4 4 \sqrt{a^2+b^2}$[/tex]
[tex]$\log 3+\log 4$[/tex]
[tex]$\log _4 3+\log _4 4$[/tex]
[tex]$\log 3 \cdot \log 4$[/tex]
Write [tex]$\log _7(2 \cdot 6)+\log _7 3$[/tex] as a single log.
[tex]$\log _7 11$[/tex]
[tex]$\log _7 15$[/tex]
[tex]$\log _7 36$[/tex]
Expand: [tex]$\log _h(9 j k)$[/tex]
[tex]$\log _h 9 \cdot \log _h j \cdot \log _h k$[/tex]
[tex]$\log _h 9+\log _h j+\log _h k$[/tex]
[tex]$\log 9+\log j+\log k$[/tex]
Answer (1)
Expand lo g 4 12 using the product property: lo g 4 12 = lo g 4 3 + lo g 4 4 .
Write lo g 7 ( 2 ⋅ 6 ) + lo g 7 3 as a single log: lo g 7 ( 2 ⋅ 6 ) + lo g 7 3 = lo g 7 36 .
Expand lo g h ( 9 jk ) using the product property: lo g h ( 9 jk ) = lo g h 9 + lo g h j + lo g h k .
Explanation
Understanding the Problem We are given the product property of logarithms: lo g b x y = lo g b x + lo g b y . We need to use this property to solve three subproblems.
Expanding log_4 12 Task 1: Expand lo g 4 12 so that it can be evaluated, given lo g 4 3 ≈ 0.792 .
We can write 12 as 3 × 4 . Therefore, using the product property of logarithms, we have lo g 4 12 = lo g 4 ( 3 × 4 ) = lo g 4 3 + lo g 4 4 Since lo g 4 4 = 1 , we get lo g 4 12 = lo g 4 3 + 1 ≈ 0.792 + 1 = 1.792 So the correct expansion is lo g 4 3 + lo g 4 4 .
Combining Logarithms Task 2: Write lo g 7 ( 2 ⋅ 6 ) + lo g 7 3 as a single log. First, simplify the expression inside the first logarithm: 2 ⋅ 6 = 12 . So, we have lo g 7 12 + lo g 7 3 .
Using the product property of logarithms, we can combine these into a single logarithm: lo g 7 12 + lo g 7 3 = lo g 7 ( 12 × 3 ) = lo g 7 36 So the single log is lo g 7 36 .
Expanding log_h (9jk) Task 3: Expand lo g h ( 9 jk ) .
Using the product property of logarithms, we have lo g h ( 9 jk ) = lo g h 9 + lo g h j + lo g h k So the expanded form is lo g h 9 + lo g h j + lo g h k .
Final Answers In summary:
lo g 4 12 = lo g 4 3 + lo g 4 4
lo g 7 ( 2 ⋅ 6 ) + lo g 7 3 = lo g 7 36
lo g h ( 9 jk ) = lo g h 9 + lo g h j + lo g h k
Examples
Logarithms are used in many scientific and engineering fields. For example, the Richter scale uses logarithms to measure the magnitude of earthquakes. If an earthquake has a magnitude of 6.0 on the Richter scale, it is ten times stronger than an earthquake with a magnitude of 5.0. Similarly, in chemistry, pH is a logarithmic scale used to measure the acidity or alkalinity of a solution. These examples show how logarithms help simplify and represent large ranges of values in a meaningful way.
