Solve the equation: [tex]$\log _4(3 x+2)-1=\log _4 8$[/tex].
Answer (1)
Add 1 to both sides: lo g 4 ( 3 x + 2 ) = lo g 4 8 + 1 .
Rewrite 1 as lo g 4 4 and combine logarithms: lo g 4 ( 3 x + 2 ) = lo g 4 ( 8 × 4 ) = lo g 4 32 .
Equate arguments: 3 x + 2 = 32 .
Solve for x : x = 3 30 = 10 .
Explanation
Isolating Logarithmic Terms We are given the equation lo g 4 ( 3 x + 2 ) − 1 = lo g 4 8 . Our goal is to solve for x . First, we need to isolate the logarithmic terms.
Adding 1 to Both Sides Add 1 to both sides of the equation: lo g 4 ( 3 x + 2 ) = lo g 4 8 + 1
Rewriting 1 as a Logarithm We can rewrite 1 as a logarithm with base 4: 1 = lo g 4 4 . Substitute this into the equation: lo g 4 ( 3 x + 2 ) = lo g 4 8 + lo g 4 4
Combining Logarithms Using the logarithm property lo g a b + lo g a c = lo g a ( b c ) , we can combine the logarithms on the right side: lo g 4 ( 3 x + 2 ) = lo g 4 ( 8 × 4 ) lo g 4 ( 3 x + 2 ) = lo g 4 32
Equating Arguments Since the logarithms are equal, their arguments must be equal: 3 x + 2 = 32
Solving for x Now, we solve the linear equation for x . Subtract 2 from both sides: 3 x = 32 − 2 3 x = 30
Final Solution Divide both sides by 3: x = 3 30 x = 10
Checking the Solution We need to check if this solution is valid by ensuring that 0"> 3 x + 2 > 0 . Substituting x = 10 , we get 0"> 3 ( 10 ) + 2 = 32 > 0 , so the solution is valid.
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 is crucial for making accurate predictions and analyses in these areas. For example, if we know the intensity of an earthquake, we can use a logarithmic equation to find its magnitude. Similarly, in finance, logarithmic scales are used to analyze investment growth and risk.
