Final answer:
To rewrite the equation e²x = 10 without logarithms, we take the natural logarithm of both sides to obtain x = (1/2)ln(10). This uses the inverse relationship between exponential functions and logarithms. Option b) x = (1/2)ln(10) is the correct answer.
Step-by-step explanation:
The question asks us to rewrite the equation e²x = 10 without using logarithms. Understanding exponents and logarithms is crucial here. The exponential and natural logarithm are indeed inverse functions, meaning that applying a logarithm to an exponentiated term simply gives us the exponent back, such as ln(eˣ) = x, and vice versa, eˆln(x) = x. Therefore, to solve for x in the given equation, we take the natural logarithm of both sides to get ln(e²x) = ln(10).
Using the property of logarithms that ln(e²x) = 2x, we can simplify our equation to 2x = ln(10). Dividing both sides by 2, we get x = (1/2) ∙ ln(10). The natural logarithm (ln) of a number is the power to which e must be raised to get the number. The constant e is approximately 2.7182818. For example, the natural logarithm of 10 is approximately 2.303, because e raised to the power of approximately 2.303 equals 10. Therefore, the correct option that represents the solution to the original equation is option b) x = (1/2)ln(10).