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

Verified step by step guidance

1. Start with the given equation: 5 ln(2x) = 20.
2. Isolate the logarithmic expression by dividing both sides of the equation by 5, resulting in ln(2x) = 4.
3. Rewrite the logarithmic equation in its equivalent exponential form using the definition of natural logarithm: if ln(a) = b, then a = e^b. So, 2x = e^4.
4. Solve for x by dividing both sides of the equation by 2: x = (e^4) / 2.
5. Check the domain restriction for the logarithmic function: the argument of ln(2x) must be greater than 0, so 2x > 0, which means x > 0. Confirm that your solution satisfies this condition.

Key Concepts

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 simplifying and solving logarithmic 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 the exact solution to logarithmic equations.