Given [tex]log _4 3 \approx 0.792[/tex] and [tex]log _4 21 \approx 2.196[/tex], what is [tex]log _4 7[/tex]?
Answer (1)
Use the logarithm property lo g a ( x y ) = lo g a x + lo g a y to express lo g 4 21 as lo g 4 3 + lo g 4 7 .
Rearrange the equation to solve for lo g 4 7 : lo g 4 7 = lo g 4 21 − lo g 4 3 .
Substitute the given values: lo g 4 7 ≈ 2.196 − 0.792 .
Calculate the result: lo g 4 7 ≈ 1.404 .
Explanation
Understanding the Problem We are given the values of lo g 4 3 and lo g 4 21 , and we want to find the value of lo g 4 7 . We can use the properties of logarithms to relate these values.
Using Logarithm Properties We know that 21 = 3 × 7 . Using the logarithm property lo g a ( x y ) = lo g a x + lo g a y , we can write lo g 4 21 = lo g 4 ( 3 × 7 ) = lo g 4 3 + lo g 4 7 .
Isolating the Desired Term Now we can solve for lo g 4 7 by rearranging the equation: lo g 4 7 = lo g 4 21 − lo g 4 3 .
Substituting Given Values We are given that lo g 4 3 ≈ 0.792 and lo g 4 21 ≈ 2.196 . Substituting these values into the equation, we get lo g 4 7 ≈ 2.196 − 0.792 .
Calculating the Result Calculating the difference, we find that lo g 4 7 ≈ 1.404 .
Final Answer Therefore, lo g 4 7 ≈ 1.404 .
Examples
Logarithms are used in many scientific and engineering applications, such as measuring the intensity of earthquakes (the Richter scale) or the loudness of sound (decibels). In computer science, logarithms are used to analyze the efficiency of algorithms. For example, if you know how the logarithm of a value changes, you can predict how the value itself will change, which is useful in various fields like data analysis and cryptography. Understanding logarithmic relationships helps in making informed decisions based on data trends and patterns.
