What is the value of $\log 20 \cdot \log 5$?
Answer (1)
Rewrite lo g 20 as lo g 2 + 1 and lo g 5 as 1 − lo g 2 .
Multiply the expressions: ( lo g 2 + 1 ) ( 1 − lo g 2 ) = 1 − ( lo g 2 ) 2 .
Calculate the approximate value: 1 − ( lo g 2 ) 2 ≈ 1 − ( 0.30103 ) 2 ≈ 0.90938 .
The value of lo g 20 \t ⋅ lo g 5 is approximately 0.90938, which is not among the given options.
Explanation
Understanding the Problem We are asked to find the value of the expression lo g 20 \t ⋅ lo g 5 . The base of the logarithm is not specified, so we assume it is base 10.
Rewriting the Logarithms We can rewrite lo g 20 as lo g ( 4 ⋅ 5 ) or lo g ( 2 ⋅ 10 ) or lo g ( 2 ⋅ 2 ⋅ 5 ) . We can rewrite lo g 5 as lo g ( 10/2 ) .
Expressing in terms of log 2 lo g 20 = lo g ( 2 ⋅ 10 ) = lo g 2 + lo g 10 = lo g 2 + 1 . Also, lo g 5 = lo g ( 10/2 ) = lo g 10 − lo g 2 = 1 − lo g 2 .
Multiplying the Expressions Then lo g 20 ⋅ lo g 5 = ( lo g 2 + 1 ) ( 1 − lo g 2 ) = 1 − ( lo g 2 ) 2 .
Calculating the Value Using a calculator, we find that lo g 2 ≈ 0.30103 , so ( lo g 2 ) 2 ≈ 0.09062 . Therefore, 1 − ( lo g 2 ) 2 ≈ 1 − 0.09062 = 0.90938 .
Alternative Calculation Alternatively, lo g 20 ⋅ lo g 5 = lo g ( 4 × 5 ) ⋅ lo g ( 5 ) = ( lo g 4 + lo g 5 ) ⋅ lo g 5 = ( 2 lo g 2 + lo g 5 ) ⋅ lo g 5 = 2 lo g 2 lo g 5 + ( lo g 5 ) 2 = 2 lo g 2 ( 1 − lo g 2 ) + ( 1 − lo g 2 ) 2 = 2 lo g 2 − 2 ( lo g 2 ) 2 + 1 − 2 lo g 2 + ( lo g 2 ) 2 = 1 − ( lo g 2 ) 2 ≈ 0.90938 .
Final Answer The value of lo g 20 ⋅ lo g 5 = 1 − ( lo g 2 ) 2 ≈ 0.90938 . Among the given choices, the closest value is none of them. However, if we consider the expression 1 − ( lo g 2 ) 2 , we can see that it is close to 1.
Selecting the Correct Option Since lo g 20 \t ⋅ lo g 5 = 1 − ( lo g 2 ) 2 ≈ 0.90938 , the closest answer among the options is none of them. However, the calculation gives approximately 0.90938.
Examples
Logarithms are used in many scientific and engineering fields. For example, the Richter scale uses logarithms to measure the magnitude of earthquakes. The decibel scale uses logarithms to measure the intensity of sound. In chemistry, pH is a logarithmic scale used to measure the acidity or basicity of a solution. Understanding logarithmic properties helps in simplifying calculations and interpreting data in these fields.
