Solve.
Write your answer in simplest form using integers, fractions, and natural logarithms.
[tex]$e^x=5$[/tex]
[tex]$x=\square$[/tex]
Answer (2)
Take the natural logarithm of both sides of the equation e x = 5 .
Apply the property ln ( e x ) = x to simplify the left side.
The solution is x = ln ( 5 ) .
The final answer is ln ( 5 ) .
Explanation
Understanding the Problem We are given the equation e x = 5 and asked to solve for x . The solution should be in simplest form using integers, fractions, and natural logarithms.
Applying Natural Logarithm To solve for x , we take the natural logarithm of both sides of the equation:
Taking ln on both sides ln ( e x ) = ln ( 5 )
Simplifying the equation Using the property that ln ( e x ) = x , we simplify the left side of the equation:
The solution x = ln ( 5 )
Final Answer Since 5 is an integer, and the problem asks for the answer in terms of natural logarithms, the solution is simply ln ( 5 ) .
Examples
The natural logarithm is used to solve exponential equations, which appear in various fields such as finance (calculating compound interest), physics (radioactive decay), and computer science (algorithm analysis). For instance, if you want to determine how long it takes for an investment to double at a certain interest rate, you would use logarithms to solve the exponential growth equation. Similarly, in radioactive decay, logarithms help determine the half-life of a substance.
To solve the equation e x = 5 , we take the natural logarithm of both sides, yielding x = ln ( 5 ) . Therefore, the final answer is ln ( 5 ) . This demonstrates the relationship between exponential functions and logarithms.
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