Maria wrote the equation $\log \left(\frac{x}{2}\right)+\log \left(\frac{20}{x^2}\right)=\log 8$. What is the solution to Maria's equation?
A. $x=\frac{3}{10}$
B. $x=\frac{4}{5}$
C. $x=\frac{5}{4}$
D. $x=\frac{10}{3}$
Answer (1)
Combine the logarithms using the property lo g a + lo g b = lo g ( ab ) .
Simplify the equation to lo g ( x 10 ) = lo g 8 .
Remove the logarithms and set the arguments equal: x 10 = 8 .
Solve for x : x = 4 5 .
4 5
Explanation
Understanding the Problem We are given the equation lo g ( 2 x ) + lo g ( x 2 20 ) = lo g 8 and we need to find the value of x that satisfies this equation. We will use properties of logarithms to simplify the equation and then solve for x .
Combining Logarithms Using the logarithm property lo g a + lo g b = lo g ( ab ) , we can combine the two logarithms on the left side of the equation: lo g ( 2 x ) + lo g ( x 2 20 ) = lo g ( 2 x ⋅ x 2 20 ) = lo g ( 2 x 2 20 x ) = lo g ( x 10 ) So the equation becomes: lo g ( x 10 ) = lo g 8
Removing Logarithms Since the logarithms are equal, their arguments must be equal as well: x 10 = 8
Solving for x Now we solve for x :
10 = 8 x x = 8 10 = 4 5
Final Answer Therefore, the solution to Maria's equation is x = 4 5 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. For example, if we want to determine how long it takes for an investment to double at a certain interest rate, we can use logarithms to solve the compound interest formula. Understanding logarithmic equations helps in making informed decisions in finance, science, and engineering.
