Solve for x.
[tex]\ln x=8[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution is [tex]$x=$[/tex] [ ] .
(Type an exact answer in simplified form. Type an expression using logarithms.)
B. The solution is not a real number.
Answer (2)
Apply the exponential function to both sides of the equation: e l n x = e 8 .
Simplify using the property e l n x = x : x = e 8 .
The solution is x = e 8 .
Therefore, the final answer is e 8 .
Explanation
Understanding the Problem We are given the equation ln x = 8 and we want to solve for x . The natural logarithm, denoted as ln x , is the logarithm to the base e , where e is an irrational number approximately equal to 2.71828. To solve for x , we need to undo the natural logarithm.
Applying the Exponential Function To isolate x , we can use the exponential function with base e . The exponential function is the inverse of the natural logarithm function. This means that if we raise e to the power of both sides of the equation, we can eliminate the natural logarithm. So, we have: e l n x = e 8
Simplifying the Equation Using the property that e l n x = x , the equation simplifies to: x = e 8
Finding the Solution The value of e 8 is approximately 2980.957987. Therefore, the solution for x is e 8 .
x = e 8 ≈ 2980.957987
Examples
Logarithmic equations like the one we solved are useful in various fields. For instance, they can model the growth of populations, the decay of radioactive substances, and the calculation of sound intensity (decibels). Understanding how to solve these equations allows us to predict future population sizes, determine the age of artifacts using carbon dating, or design audio equipment effectively. The ability to manipulate and solve logarithmic equations provides a powerful tool for analyzing and understanding real-world phenomena.
To solve the equation ln x = 8 , we apply the exponential function, yielding the solution x = e 8 . This can be approximated numerically as about 2980.96 but is best expressed in terms of e for exactness.
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