Use the change of base formula to compute $\log _7 4$. Round your answer to the nearest thousandth.
Answer (2)
Apply the change of base formula: lo g 7 4 = l o g 7 l o g 4 .
Calculate the values of lo g 4 and lo g 7 : lo g 4 ≈ 0.60206 , lo g 7 ≈ 0.845098 .
Divide l o g 7 l o g 4 : 0.845098 0.60206 ≈ 0.712414 .
Round the result to the nearest thousandth: 0.712 .
Explanation
Understanding the Problem and Change of Base Formula We are asked to compute lo g 7 4 using the change of base formula and round the answer to the nearest thousandth. The change of base formula states that lo g a b = l o g c a l o g c b for any positive a , b , c where a = 1 and c = 1 . We will use the common logarithm (base 10) for our calculation.
Applying the Change of Base Formula Applying the change of base formula, we have: lo g 7 4 = lo g 10 7 lo g 10 4
Calculating the Logarithms Using a calculator, we find that lo g 10 4 ≈ 0.60206 and lo g 10 7 ≈ 0.845098 . Therefore, lo g 7 4 = 0.845098 0.60206 ≈ 0.712414
Rounding to the Nearest Thousandth Rounding the result to the nearest thousandth, we get 0.712.
Final Answer Therefore, lo g 7 4 ≈ 0.712 .
Examples
The change of base formula is useful in many real-world applications, such as calculating the intensity of earthquakes on the Richter scale or determining the acidity (pH) of a solution. For example, if we want to compare the relative acidity of two solutions with different hydrogen ion concentrations, we can use logarithms with a common base to easily compare their pH values. This allows scientists to quickly assess and compare the properties of different substances.
Using the change of base formula, lo g 7 4 is calculated as l o g 10 7 l o g 10 4 , which results in approximately 0.712 when rounded to the nearest thousandth.
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