Solve the logarithmic equation. Be sure to reject any value of [tex]$x$[/tex] that is not in the domain of the original logarithmic expression. [tex]$7 \ln (7 x)=28$[/tex]
Answer (2)
Divide both sides of the equation by 7: ln ( 7 x ) = 4 .
Exponentiate both sides using base e : e l n ( 7 x ) = e 4 .
Simplify: 7 x = e 4 .
Solve for x : x = 7 e 4 .
Explanation
Understanding the Problem We are given the logarithmic equation 7 ln ( 7 x ) = 28 . Our goal is to solve for x , making sure that our solution is within the domain of the original logarithmic expression. The domain of ln ( 7 x ) is 0"> 7 x > 0 , which means 0"> x > 0 .
Isolating the Logarithm First, we divide both sides of the equation by 7 to isolate the natural logarithm: ln ( 7 x ) = 7 28 = 4
Exponentiating Both Sides Next, we exponentiate both sides of the equation using the base e to remove the natural logarithm: e l n ( 7 x ) = e 4
Simplifying the Equation Using the property e l n ( a ) = a , we simplify the left side of the equation: 7 x = e 4
Solving for x Now, we divide both sides of the equation by 7 to solve for x : x = 7 e 4
Final Answer We need to check if the solution is in the domain of the original logarithmic expression, which requires 0"> x > 0 . Since 0"> e 4 > 0 , then 0"> 7 e 4 > 0 , so the solution is valid. We can approximate the value of e 4 as 54.598, so x = 7 54.598 ≈ 7.7997 . Therefore, the solution to the equation is: x = 7 e 4
Examples
Logarithmic equations are used in various fields such as physics, engineering, and finance. For example, in radioactive decay, the amount of a radioactive substance remaining after time t is given by N ( t ) = N 0 e − k t , where N 0 is the initial amount and k is the decay constant. If we want to find the time it takes for the substance to decay to a certain level, we need to solve a logarithmic equation. Similarly, in finance, logarithmic equations are used to calculate the time it takes for an investment to double at a given interest rate.
To solve 7 ln ( 7 x ) = 28 , we isolate the logarithm and exponentiate both sides, yielding x = 7 e 4 . This solution is valid since it meets the domain condition 0"> x > 0 . Therefore, the solution to the equation is x = 7 e 4 .
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