\frac{\log 125-6 \log 25}{\log 625+\frac{1}{2} \log 25}
Answer (2)
To simplify the expression l o g 625 + 2 1 l o g 25 l o g 125 − 6 l o g 25 , we rewrite the arguments as powers of 5, apply the power rule of logarithms, and simplify. The final result is − 5 9 .
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Rewrite the expression using the property a = b c , where a is the argument of the logarithm and c is the exponent.
Apply the power rule of logarithms: lo g a b = b lo g a .
Simplify the expression by combining like terms.
Cancel out the common logarithmic term to obtain the final result: − 5 9 .
Explanation
Understanding the Problem and Strategy We are asked to simplify the expression l o g 625 + 2 1 l o g 25 l o g 125 − 6 l o g 25 . To do this, we will use properties of logarithms to rewrite the expression in a simpler form. First, we express the arguments of the logarithms as powers of 5: 125 = 5 3 , 25 = 5 2 , and 625 = 5 4 .
Applying the Power Rule of Logarithms Now we rewrite the expression using the powers of 5: lo g 5 4 + 2 1 lo g 5 2 lo g 5 3 − 6 lo g 5 2 Next, we apply the power rule of logarithms, which states that lo g a b = b lo g a . Applying this rule to each term in the expression, we get: 4 lo g 5 + 2 1 ( 2 lo g 5 ) 3 lo g 5 − 6 ( 2 lo g 5 )
Simplifying the Expression Now we simplify the expression: 4 lo g 5 + lo g 5 3 lo g 5 − 12 lo g 5 Combining the terms in the numerator and the denominator, we have: 5 lo g 5 − 9 lo g 5
Final Simplification and Answer Finally, we cancel out the lo g 5 terms, since lo g 5 = 0 :
5 − 9 Thus, the simplified value of the expression is − 5 9 .
Examples
Logarithms are used in many scientific and engineering fields. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. Similarly, the pH scale, which measures the acidity or alkalinity of a solution, is also a logarithmic scale. Understanding how to manipulate logarithmic expressions can help in analyzing data and solving problems in these fields. For instance, if you are comparing the intensities of two earthquakes, you can use the properties of logarithms to determine how much stronger one earthquake is compared to the other. If one earthquake measures 6 on the Richter scale and another measures 4, the first earthquake is 1 0 6 − 4 = 1 0 2 = 100 times stronger than the second.
