Find the logarithm using the change of base formula. [tex]$\log _4 20$[/tex]
Answer (2)
Apply the change of base formula to express the logarithm in terms of a common base, such as base 10 or base e .
Use the formula lo g a b = l o g c a l o g c b .
Calculate the logarithm using a calculator.
The final answer is: 2.1609640474436813 .
Explanation
Understanding the Change of Base Formula We are asked to find the value of lo g 4 20 using the change of base formula. The change of base formula allows us to convert a logarithm from one base to another. The formula is given by: lo g a b = lo g c a lo g c b where a , b , and c are positive numbers and a , c = 1 .
Applying the Change of Base Formula We can use the change of base formula to convert the given logarithm to base 10. Using a = 4 , b = 20 , and c = 10 , we have: lo g 4 20 = lo g 10 4 lo g 10 20 We can also use the natural logarithm (base e ) for the change of base. Using a = 4 , b = 20 , and c = e , we have: lo g 4 20 = ln 4 ln 20
Calculating the Logarithm Now, we can use a calculator to find the values of lo g 10 20 and lo g 10 4 , or ln 20 and ln 4 , and then divide them. Alternatively, we can use the property of logarithms that lo g ( x y ) = lo g x + lo g y to simplify the expression. Since 20 = 4 × 5 , we have lo g 4 20 = lo g 4 ( 4 × 5 ) = lo g 4 4 + lo g 4 5 = 1 + lo g 4 5 . Thus, the answer can also be expressed as 1 + lo g 4 5 .
Using a calculator, we find that lo g 4 20 ≈ 2.160964 .
Final Answer Using a calculator to evaluate the logarithm directly, we get: lo g 4 20 ≈ 2.1609640474436813 Therefore, the value of lo g 4 20 is approximately 2.160964.
Examples
Logarithms are used in many scientific fields, such as physics, chemistry, and engineering. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. The pH scale, which measures the acidity or alkalinity of a solution, is also a logarithmic scale. Logarithms are also used in computer science to analyze the efficiency of algorithms. The change of base formula is particularly useful when a calculator only has logarithms for certain bases (like base 10 or base e), but you need to calculate a logarithm with a different base. By using the change of base formula, you can easily convert the logarithm to a base that your calculator can handle.
To find lo g 4 20 , we use the change of base formula: lo g 4 20 = l o g 10 4 l o g 10 20 . Calculating this gives an approximate value of 2.160964 .
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