Solve. Write your answer in simplest form using integers, fractions, and natural logarithms.
[tex]\begin{array}{l}
e^x=7 \\nx=\frac{1}{\frac{\square}{\square} \ln }\end{array}[/tex]
Answer (2)
Take the natural logarithm of both sides of the equation e x = 7 to get ln ( e x ) = ln ( 7 ) .
Simplify the left side using the property ln ( e x ) = x , so x = ln ( 7 ) .
Rewrite the solution in the desired form x = □ □ l n ( □ ) 1 .
The final answer is x = ln ( 7 ) , which can be written as x = l n ( 7 ) 1 1 , so the missing fraction is 1 1 and the missing value inside the logarithm is 7. Thus, x = ln ( 7 ) .
Explanation
Understanding the Problem We are given the equation e x = 7 and asked to solve for x , expressing the answer in terms of natural logarithms and a fraction.
Taking the Natural Logarithm To solve for x , we take the natural logarithm of both sides of the equation: ln ( e x ) = ln ( 7 ) Using the property that ln ( e x ) = x , we have: x = ln ( 7 )
Rewriting the Solution Now, we need to express the solution in the form x = □ □ l n ( □ ) 1 . Since x = ln ( 7 ) , we can rewrite it as: x = l n ( 7 ) 1 1 Thus, the missing fraction is 1 1 = 1 and the missing value inside the logarithm is 7. So, we have: x = l n ( 7 ) 1 1
Final Answer Therefore, the solution is x = ln ( 7 ) .
Examples
Natural logarithms are used in various fields such as physics, engineering, and finance. For example, they are used to model exponential decay in radioactive materials, calculate growth rates in populations, and determine the time it takes for an investment to double at a continuous interest rate. Understanding how to solve equations involving natural logarithms is crucial in these applications.
To solve e x = 7 , we take the natural logarithm, resulting in x = ln ( 7 ) . This can be expressed as x = l n ( 7 ) 1 1 with missing values as 1 1 and 7. Therefore, the complete answer is x = ln ( 7 ) .
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