Solve the equation [tex]e^{x-2}=8[/tex] and express the
B) [tex] \{\ln 10\} [/tex]
C) [tex] \{-2 [/tex]
Answer (2)
Take the natural logarithm of both sides: ln ( e x − 2 ) = ln ( 8 ) .
Simplify using logarithm properties: x − 2 = ln ( 8 ) .
Isolate x : x = ln ( 8 ) + 2 .
Express the solution: x = ln ( 8 e 2 ) .
The final answer is ln ( 8 e 2 ) .
Explanation
Understanding the Problem We are given the equation e x − 2 = 8 and asked to solve for x .
Taking the Natural Logarithm To solve for x , we first take the natural logarithm of both sides of the equation: ln ( e x − 2 ) = ln ( 8 ) Using the property that ln ( e u ) = u , we have: x − 2 = ln ( 8 ) Now, we isolate x by adding 2 to both sides: x = ln ( 8 ) + 2
Simplifying the Expression We can rewrite 2 as ln ( e 2 ) , so we have: x = ln ( 8 ) + ln ( e 2 ) Using the property that ln ( a ) + ln ( b ) = ln ( ab ) , we can combine the logarithms: x = ln ( 8 e 2 ) Since e 2 ≈ 7.389 , we have 8 e 2 ≈ 8 × 7.389 = 59.112 . Thus, x = ln ( 59.112 ) ≈ 4.079 .
Final Answer Therefore, the solution to the equation e x − 2 = 8 is x = ln ( 8 e 2 ) .
Examples
Exponential equations are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that compounds interest continuously, the amount of money you have after a certain time can be modeled by an exponential equation. Solving such equations allows you to determine how long it will take for your investment to reach a certain value.
To solve the equation e x − 2 = 8 , we take the natural logarithm of both sides, simplify, and isolate x to find the solution is x = ln ( 8 e 2 ) .
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