Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 5 ln(2x)=20

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Start with the given equation: \(5 \ln(2x) = 20\).

Isolate the logarithmic expression by dividing both sides of the equation by 5: \(\ln(2x) = \frac{20}{5}\).

Simplify the right side: \(\ln(2x) = 4\).

Rewrite the logarithmic equation in its exponential form using the fact that \(\ln(a) = b\) means \(a = e^b\): \(2x = e^4\).

Solve for \(x\) by dividing both sides by 2: \(x = \frac{e^4}{2}\). Remember to check that this value of \(x\) keeps the argument of the logarithm positive, which it does since \(e^4 > 0\).

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주요 개념

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

Properties of Logarithms

Understanding the properties of logarithms, such as the ability to rewrite equations involving logarithms and exponents, is essential. For example, the natural logarithm ln(a^b) can be expressed as b ln(a), which helps in isolating variables and simplifying equations.

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive arguments because the logarithm of zero or a negative number is undefined. When solving logarithmic equations, it is crucial to check that the solutions keep the argument inside the logarithm positive.

Solving Exponential Equations

After isolating the logarithmic expression, converting it to its equivalent exponential form allows solving for the variable. For example, if ln(y) = c, then y = e^c. This step is key to finding exact solutions before approximating decimal values.