Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e^x=5.7

검증된 단계별 안내

Identify the given exponential equation: \(e^{x} = 5.7\).

Recall that the natural logarithm function \(\ln(x)\) is the inverse of the exponential function with base \(e\). This means applying \(\ln\) to both sides will help isolate \(x\).

Apply the natural logarithm to both sides of the equation: \(\ln(e^{x}) = \ln(5.7)\).

Use the logarithmic identity \(\ln(e^{x}) = x\) to simplify the left side, resulting in \(x = \ln(5.7)\).

To find a decimal approximation, use a calculator to evaluate \(\ln(5.7)\) and round the result to two decimal places.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Functions

An exponential function has the form f(x) = a^x, where the variable is in the exponent. In this problem, the base is the constant e (Euler's number, approximately 2.718), making it a natural exponential function. Understanding how to work with exponential functions is essential to isolate the variable in the exponent.

Natural Logarithms

The natural logarithm, denoted ln(x), is the inverse function of the exponential function with base e. Applying the natural logarithm to both sides of an equation like e^x = 5.7 allows you to solve for x by 'undoing' the exponentiation, since ln(e^x) = x.

Using a Calculator for Approximations

After expressing the solution in terms of logarithms, a calculator is used to find a decimal approximation. This involves evaluating the logarithm (e.g., ln(5.7)) and rounding the result to the desired precision, here to two decimal places, to provide a practical numerical answer.

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ex=5.7