Use the change of base formula to compute $\log _{1 / 9} 6$. Round your answer to the nearest thousandth.
Answer (1)
Apply the change of base formula: lo g 1/9 6 = l o g ( 1/9 ) l o g 6 .
Calculate the values of lo g 6 and lo g ( 1/9 ) : lo g 6 ≈ 0.77815 and lo g ( 1/9 ) ≈ − 0.95424 .
Divide lo g 6 by lo g ( 1/9 ) : − 0.95424 0.77815 ≈ − 0.81546 .
Round the result to the nearest thousandth: − 0.815 .
Explanation
Understanding the Problem We are asked to compute lo g 1/9 6 using the change of base formula and round the result to the nearest thousandth.
Change of Base Formula 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 , and c where a = 1 and c = 1 . We will use the common logarithm (base 10) for our change of base.
Applying the Formula Applying the change of base formula, we have lo g 1/9 6 = lo g 10 ( 1/9 ) lo g 10 6 .
Calculating Logarithms We find that lo g 10 6 ≈ 0.77815 and lo g 10 ( 1/9 ) ≈ − 0.95424 .
Dividing the Logarithms Dividing these values, we get lo g 10 ( 1/9 ) lo g 10 6 ≈ − 0.95424 0.77815 ≈ − 0.81546.
Rounding the Result Rounding to the nearest thousandth, we get − 0.815 .
Final Answer Therefore, lo g 1/9 6 ≈ − 0.815 .
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 pH of a solution in chemistry. 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 for a standardized and easily interpretable comparison.
