Which expression is equivalent to [tex]$\log 18-\log (p+2)$[/tex]?
A. [tex]$\log \frac{p+2}{18}$[/tex]
B. [tex]$\log \frac{18}{p+2}$[/tex]
C. [tex]$\log \frac{20}{p}$[/tex]
D. [tex]$\log [18 \cdot(p+2)]$[/tex]
Answer (1)
Apply the logarithm property: lo g a − lo g b = lo g b a .
Substitute a = 18 and b = p + 2 into the property.
Simplify the expression: lo g 18 − lo g ( p + 2 ) = lo g p + 2 18 .
The equivalent expression is lo g p + 2 18 .
Explanation
Understanding the Problem We are given the expression lo g 18 − lo g ( p + 2 ) and we need to find an equivalent expression from the given options. The key property of logarithms that we'll use is that the difference of two logarithms is the logarithm of the quotient.
Applying the Logarithm Property The logarithm property states that lo g a − lo g b = lo g b a . We will apply this property to the given expression.
Substituting the Values In our case, a = 18 and b = p + 2 . Substituting these values into the logarithm property, we get: lo g 18 − lo g ( p + 2 ) = lo g p + 2 18
Comparing with the Options Now we compare our result with the given options: lo g 18 p + 2 lo g p + 2 18 lo g p 20 lo g [ 18 ⋅ ( p + 2 )] The expression equivalent to lo g 18 − lo g ( p + 2 ) is lo g p + 2 18 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. The property lo g a − lo g b = lo g b a allows us to simplify expressions and solve equations involving logarithms, making these calculations easier. For example, if we know the logarithm of the initial population and the logarithm of the current population, we can use this property to find the logarithm of the population growth factor.
