Find the value of the logarithmic function [tex]$\log _8\left(\frac{1}{8}\right)$[/tex]
Answer (2)
Rewrite the fraction as a power of the base: 8 1 = 8 − 1 .
Apply the logarithm property: lo g b ( b x ) = x .
Simplify the expression: lo g 8 ( 8 − 1 ) = − 1 .
The value of the logarithmic expression is − 1 .
Explanation
Understanding the problem We are asked to find the value of the logarithmic expression lo g 8 ( 8 1 ) . This means we need to determine what power we must raise 8 to in order to get 8 1 .
Rewriting the argument We can rewrite 8 1 as 8 − 1 . This is because 8 − 1 = 8 1 1 = 8 1 .
Applying the logarithm property Now we have lo g 8 ( 8 − 1 ) . Using the property of logarithms that lo g b ( b x ) = x , we can simplify this expression. In our case, b = 8 and x = − 1 .
Finding the value Therefore, lo g 8 ( 8 − 1 ) = − 1 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale) or the loudness of sound (decibels). They are also used in computer science to analyze the efficiency of algorithms and in finance to calculate compound interest. Understanding logarithms helps us to quantify and compare quantities that vary over a wide range of values.
The value of lo g 8 ( 8 1 ) is − 1 because 8 1 can be expressed as 8 − 1 , and applying the logarithmic property lo g b ( b x ) = x gives us the result. Thus, lo g 8 ( 8 − 1 ) = − 1 .
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