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. 6+2 ln x=5

Verified step by step guidance

1. Start with the given equation: \$6 + 2 \ln x = 5$.
2. Isolate the logarithmic term by subtracting 6 from both sides: \$2 \ln x = 5 - 6$.
3. Simplify the right side: \$2 \ln x = -1$.
4. Divide both sides by 2 to solve for \(\ln x\): \(\ln x = \frac{-1}{2}\).
5. Rewrite the equation in exponential form to solve for \(x\): \(x = e^{\frac{-1}{2}}\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the natural logarithm (ln) and its relationship to exponents, is essential. For example, ln(x) is the power to which e must be raised to get x. These properties help in isolating the logarithmic term and simplifying the equation.

Domain of Logarithmic Functions

The domain of a logarithmic function ln(x) is x > 0 because the logarithm of zero or a negative number is undefined. When solving logarithmic equations, it is crucial to check that the solutions fall within this domain to reject extraneous or invalid answers.

Solving Logarithmic Equations

Solving logarithmic equations often involves isolating the logarithm, rewriting the equation in exponential form, and then solving for the variable. After finding potential solutions, verify them against the domain restrictions to ensure they are valid.