Evaluate the logarithm. Round your answer to the nearest thousandth. [tex]2 \log _3\left(\frac{1}{52}\right) \approx[/tex]
Answer (1)
Apply the power rule of logarithms: 2 lo g 3 ( 52 1 ) = lo g 3 (( 52 1 ) 2 ) = lo g 3 ( 2704 1 ) .
Use the change of base formula: lo g 3 ( 2704 1 ) = l o g 10 ( 3 ) l o g 10 ( 2704 1 ) .
Use the property lo g ( 1/ x ) = − lo g ( x ) : l o g 10 ( 3 ) l o g 10 ( 2704 1 ) = l o g 10 ( 3 ) − l o g 10 ( 2704 ) .
Calculate and round to the nearest thousandth: l o g 10 ( 3 ) − l o g 10 ( 2704 ) ≈ − 7.193 .
Explanation
Understanding the Problem We are asked to evaluate the logarithm 2 lo g 3 ( 52 1 ) and round the answer to the nearest thousandth.
Applying the Power Rule We can use the power rule of logarithms, which states that n lo g b ( a ) = lo g b ( a n ) . Applying this rule, we have 2 lo g 3 ( 52 1 ) = lo g 3 ( ( 52 1 ) 2 ) = lo g 3 ( 5 2 2 1 ) = lo g 3 ( 2704 1 )
Using the Change of Base Formula We can use the change of base formula to convert the logarithm to base 10. The change of base formula is lo g b ( a ) = l o g c ( b ) l o g c ( a ) . Applying this formula, we have lo g 3 ( 2704 1 ) = lo g 10 ( 3 ) lo g 10 ( 2704 1 )
Using the Property of Logarithms We can use the property that lo g ( 1/ x ) = − lo g ( x ) . Applying this property, we have lo g 10 ( 3 ) lo g 10 ( 2704 1 ) = lo g 10 ( 3 ) − lo g 10 ( 2704 )
Calculating Logarithms Now we can calculate the values of lo g 10 ( 2704 ) and lo g 10 ( 3 ) .
lo g 10 ( 2704 ) ≈ 3.432 lo g 10 ( 3 ) ≈ 0.477
Dividing the Logarithms Now we can divide − lo g 10 ( 2704 ) by lo g 10 ( 3 ) .
lo g 10 ( 3 ) − lo g 10 ( 2704 ) ≈ 0.477 − 3.432 ≈ − 7.193
Final Answer Rounding the result to the nearest thousandth, we get − 7.193 .
Examples
Logarithms are used in many scientific fields such as chemistry, physics, and engineering. For example, the pH scale used to measure the acidity or alkalinity of a solution is based on logarithms. The Richter scale used to measure the magnitude of earthquakes is also based on logarithms. In finance, logarithms are used to calculate the time it takes for an investment to double at a given interest rate. Logarithms are also used in computer science to analyze the complexity of algorithms.
