What is the solution to [tex]$\log _2(9 x)-\log _2 3=3$[/tex]?
A. [tex]$x=\frac{3}{8}$[/tex]
B. [tex]$x=\frac{8}{3}$[/tex]
C. [tex]$x=3$[/tex]
D. [tex]$x=9$[/tex]
Answer (1)
Use the logarithm property to combine the terms: lo g 2 ( 9 x ) − lo g 2 ( 3 ) = lo g 2 ( 3 9 x ) = lo g 2 ( 3 x ) .
Rewrite the equation: lo g 2 ( 3 x ) = 3 .
Convert to exponential form: 3 x = 2 3 = 8 .
Solve for x : x = 3 8 .
The solution is 3 8 .
Explanation
Understanding the Problem We are given the equation lo g 2 ( 9 x ) − lo g 2 3 = 3 and asked to solve for x .
Applying Logarithm Properties We can use the logarithm property lo g a b − lo g a c = lo g a c b to rewrite the left side of the equation: lo g 2 ( 9 x ) − lo g 2 3 = lo g 2 3 9 x = lo g 2 ( 3 x ) So the equation becomes lo g 2 ( 3 x ) = 3 .
Converting to Exponential Form Now, we convert the logarithmic equation to an exponential equation. The equation lo g 2 ( 3 x ) = 3 means that 2 3 = 3 x .
Simplifying the Equation We simplify the left side: 2 3 = 8 , so we have 3 x = 8 .
Solving for x Finally, we solve for x by dividing both sides by 3: x = 3 8
Final Answer Therefore, the solution to the equation lo g 2 ( 9 x ) − lo g 2 3 = 3 is x = 3 8 .
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. Understanding how to solve logarithmic equations allows us to analyze and interpret data in these real-world scenarios. For example, if we know the intensity of an earthquake, we can use logarithms to find its magnitude.
