Choose the single logarithmic expression that is equivalent to the one shown.
[tex]\log 8+\log 3[/tex]
Answer (1)
∙ Use the logarithmic property lo g a + lo g b = lo g ( a × b ) .
∙ Apply the property to the given expression: lo g 8 + lo g 3 = lo g ( 8 × 3 ) .
∙ Calculate the product: 8 × 3 = 24 .
∙ The equivalent expression is lo g 24 .
Explanation
Understanding the Problem We are given the expression lo g 8 + lo g 3 and asked to find an equivalent single logarithmic expression from the choices provided.
Applying Logarithmic Properties To solve this, we will use the logarithmic property that states the sum of the logarithms of two numbers is equal to the logarithm of their product: lo g a + lo g b = lo g ( a × b )
Calculating the Product Applying this property to our expression, we have: lo g 8 + lo g 3 = lo g ( 8 × 3 ) Now, we multiply 8 and 3: 8 × 3 = 24 Therefore, lo g 8 + lo g 3 = lo g 24
Finding the Equivalent Expression Comparing our result, lo g 24 , with the given choices, we find that it matches one of the options.
Final Answer The single logarithmic expression equivalent to lo g 8 + lo g 3 is lo g 24 .
Examples
Logarithms are very useful in many real-world applications, such as calculating the magnitude of earthquakes using the Richter scale or determining the pH of a chemical solution. In finance, logarithms can help in calculating the time it takes for an investment to double at a certain interest rate. For example, if you invest $1000 at an annual interest rate, you can use logarithms to find out how long it will take for your investment to reach $2000.
