Solve for $x$ in: $\\log (7 x-3)+2 \\log 5=2+\\log (x+3)$
Answer (2)
Use logarithm properties to simplify the equation: lo g ( 7 x − 3 ) + lo g 25 = lo g 100 + lo g ( x + 3 ) .
Combine logarithms: lo g ( 25 ( 7 x − 3 )) = lo g ( 100 ( x + 3 )) .
Equate arguments: 25 ( 7 x − 3 ) = 100 ( x + 3 ) .
Solve for x : x = 5 .
Explanation
Problem Analysis and Strategy We are given the equation lo g ( 7 x − 3 ) + 2 lo g 5 = 2 + lo g ( x + 3 ) and we want to solve for x . First, we need to use logarithm properties to simplify the equation.
Applying Logarithm Properties Using the logarithm property n lo g a = lo g a n , we can rewrite the term 2 lo g 5 as lo g 5 2 = lo g 25 . Also, we can rewrite 2 as 2 lo g 10 = lo g 1 0 2 = lo g 100 . Thus, the equation becomes lo g ( 7 x − 3 ) + lo g 25 = lo g 100 + lo g ( x + 3 )
Combining Logarithms Using the logarithm property lo g a + lo g b = lo g ( ab ) , we can combine the logarithms on both sides of the equation: lo g ( 25 ( 7 x − 3 )) = lo g ( 100 ( x + 3 ))
Equating Arguments Since the logarithms are equal, their arguments must be equal. Therefore, 25 ( 7 x − 3 ) = 100 ( x + 3 ) Expanding both sides, we get 175 x − 75 = 100 x + 300
Solving for x Now, we solve for x . Subtract 100 x from both sides: 175 x − 100 x − 75 = 100 x − 100 x + 300 75 x − 75 = 300 Add 75 to both sides: 75 x − 75 + 75 = 300 + 75 75 x = 375 Divide both sides by 75: x = 75 375 = 5
Checking the Solution We need to check if the solution x = 5 is valid. The domain of the logarithms requires 0"> 7 x − 3 > 0 and 0"> x + 3 > 0 . If x = 5 , then 0"> 7 x − 3 = 7 ( 5 ) − 3 = 35 − 3 = 32 > 0 and 0"> x + 3 = 5 + 3 = 8 > 0 . Thus, x = 5 is a valid solution.
Final Answer Therefore, the solution to the equation lo g ( 7 x − 3 ) + 2 lo g 5 = 2 + lo g ( x + 3 ) is x = 5 .
Examples
Logarithmic equations are used in various fields such as finance, physics, and engineering. For example, in finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. In physics, they are used to describe the decay of radioactive materials. Understanding how to solve logarithmic equations is crucial for making informed decisions in these areas. For instance, if you want to know how long it will take for your investment to double at a 7% annual interest rate, you would use a logarithmic equation.
To solve the equation, use logarithm properties to combine and equate the expressions. This leads to the simplified form that allows solving for x , resulting in x = 5 . Validation of the solution ensures it is within the defined domain of the logarithms.
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