Use the properties of logarithms to evaluate each of the following expressions.
(a) $4 \ln e^2+\ln e^{-12}=$ $\square$
(b) $\log _3 39-\log _3 13=$ $\square$
Answer (1)
In the first expression, apply the property ln e x = x to simplify 4 ln e 2 + ln e − 12 to 4 ( 2 ) + ( − 12 ) .
Calculate 4 ( 2 ) + ( − 12 ) = 8 − 12 = − 4 .
In the second expression, use the quotient rule lo g b x − lo g b y = lo g b y x to rewrite lo g 3 39 − lo g 3 13 as lo g 3 13 39 = lo g 3 3 .
Since lo g 3 3 = 1 , the final answers are − 4 and 1 .
Explanation
Problem Analysis We are asked to evaluate two logarithmic expressions using properties of logarithms. Let's tackle them one by one.
Evaluating the first expression (a) We have the expression 4 ln e 2 + ln e − 12 .
We know that ln e x = x . Therefore, ln e 2 = 2 and ln e − 12 = − 12 .
Substituting these values into the expression, we get:
4 ln e 2 + ln e − 12 = 4 ( 2 ) + ( − 12 ) = 8 − 12 = − 4 .
Evaluating the second expression (b) We have the expression lo g 3 39 − lo g 3 13 .
We can use the quotient rule of logarithms, which states that lo g b x − lo g b y = lo g b y x .
Applying this rule, we get:
lo g 3 39 − lo g 3 13 = lo g 3 13 39 = lo g 3 3 .
Since lo g b b = 1 , we have lo g 3 3 = 1 .
Final Answer Therefore, the values of the expressions are:
(a) 4 ln e 2 + ln e − 12 = − 4
(b) lo g 3 39 − lo g 3 13 = 1
Examples
Logarithms are incredibly useful in many real-world situations, especially when dealing with exponential growth or decay. For instance, calculating the time it takes for an investment to double at a certain interest rate involves logarithms. Similarly, in chemistry, logarithms are used to measure the acidity or alkalinity of a solution using the pH scale. Understanding logarithmic properties allows us to simplify complex calculations in finance, science, and engineering, making them an essential tool for problem-solving.
