Solve the logarithmic equation.
[tex]8+2 \ln x=22[/tex]
[tex]x=[/tex]
(Simplify your answer. Type an exact answer, using [tex]e[/tex] as needed.)
Answer (2)
Subtract 8 from both sides: 2 ln x = 14 .
Divide both sides by 2: ln x = 7 .
Exponentiate both sides with base e : e l n x = e 7 .
Simplify to find the solution: x = e 7 .
Explanation
Problem Analysis We are given the equation 8 + 2 ln x = 22 and we want to solve for x .
Isolating the Logarithmic Term First, we subtract 8 from both sides of the equation: 8 + 2 ln x − 8 = 22 − 8
Simplifying the Equation This simplifies to: 2 ln x = 14
Dividing by 2 Next, we divide both sides by 2: 2 2 ln x = 2 14
Isolating the Natural Logarithm This simplifies to: ln x = 7
Exponentiating Both Sides To solve for x , we exponentiate both sides of the equation using the base e :
e l n x = e 7
Solving for x Since e l n x = x , we have: x = e 7
Examples
Logarithmic equations are useful in many fields, including finance, physics, and computer science. For example, they can be used to model the growth of investments, the decay of radioactive materials, and the complexity of algorithms. Understanding how to solve logarithmic equations is therefore a valuable skill for anyone working in these fields. Imagine you deposit money in a bank account that pays continuously compounded interest. The equation A = P e r t relates the amount of money A in the account after time t , where P is the initial principal and r is the interest rate. If you want to know how long it will take for your investment to double, you would need to solve a logarithmic equation.
The solution to the equation 8 + 2 ln x = 22 is x = e 7 . This was found by isolating the logarithmic term and exponentiating both sides. Understanding how to manipulate logarithmic equations is valuable in many fields.
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