Solve the following equation algebraically and check graphically.
[tex]8964=e^{3 x}[/tex]
[tex]x \approx[/tex] [ ]
(Do not round until the final answer. Then round to three decimal places as needed.)
Answer (2)
Take the natural logarithm of both sides: ln ( 8964 ) = ln ( e 3 x ) .
Simplify using logarithm properties: ln ( 8964 ) = 3 x .
Isolate x : x = 3 l n ( 8964 ) .
Calculate and round to three decimal places: x ≈ 3.034 .
Explanation
Problem Analysis We are given the equation 8964 = e 3 x and we want to solve for x . We will use logarithms to isolate x .
Taking the Natural Logarithm First, take the natural logarithm of both sides of the equation: ln ( 8964 ) = ln ( e 3 x )
Simplifying the Equation Using the property of logarithms, we can simplify the right side of the equation: ln ( 8964 ) = 3 x
Isolating x Now, divide both sides by 3 to isolate x : x = 3 ln ( 8964 )
Calculating x Using a calculator, we find that: x ≈ 3.0336572783069395
Rounding the Answer Rounding to three decimal places, we get: x ≈ 3.034
Examples
Exponential equations like this are used in various fields, such as calculating the growth of bacteria, the decay of radioactive substances, or the compound interest on an investment. For example, if you invest money in an account that compounds continuously, the amount of money you have after a certain time can be modeled by an exponential equation. Solving for the time it takes to reach a certain amount involves solving an equation similar to the one above.
To solve 8964 = e 3 x , we take the natural logarithm of both sides and simplify. This leads to x = 3 l n ( 8964 ) , which calculates to approximately 3.034 after rounding. Thus, the solution is x ≈ 3.034 .
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