Use logarithms to solve the exponential equation.
[tex]e^{5 x}=6[/tex]
The solution is [tex]x =[/tex] $\square$ .
(Type an integer or decimal rounded to three decimal places as needed.)
Answer (2)
To solve the equation e 5 x = 6 , we take the natural logarithm of both sides, simplify, isolate x , and find x ≈ 0.358 .
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Take the natural logarithm of both sides: ln ( e 5 x ) = ln ( 6 ) .
Simplify using the logarithm property: 5 x = ln ( 6 ) .
Isolate x : x = 5 l n ( 6 ) .
Calculate the approximate value: x ≈ 0.358 . The solution is 0.358 .
Explanation
Understanding the Problem We are given the exponential equation e 5 x = 6 and we want to solve for x using logarithms.
Applying Natural Logarithm To solve for x , we can take the natural logarithm (ln) of both sides of the equation. This gives us: ln ( e 5 x ) = ln ( 6 ) .
Simplifying the Equation Using the property of logarithms that ln ( e u ) = u , we can simplify the left side of the equation: 5 x = ln ( 6 ) .
Isolating x Now, we can isolate x by dividing both sides of the equation by 5: x = 5 ln ( 6 ) .
Calculating the Approximate Value To find the approximate value of x , we can calculate 5 l n ( 6 ) . The result of this calculation, rounded to three decimal places, is approximately 0.358.
Final Answer Therefore, the solution to the equation e 5 x = 6 is approximately x = 0.358 .
Examples
Exponential equations are used in various fields such as finance, biology, and physics. For example, they can model population growth, radioactive decay, and compound interest. In finance, understanding exponential growth helps in calculating investment returns over time. If you invest 1000 a t anann u a l in t eres t r a t eo f 5 A = 1000e^{0.05t}$. Solving such equations helps determine how long it takes for the investment to reach a certain value.
