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 by isolating the logarithmic term. Subtract 6 from both sides of the equation: $2\ln x=5-6$.
2. Simplify the right side: $2\ln x=-1$.
3. Divide both sides by 2 to solve for $\ln x$: $\ln x=\frac{-1}{2}$.
4. To solve for $x$, rewrite the equation in exponential form: $x=e^{\frac{-1}{2}}$.
5. Check the domain of the original logarithmic expression. Since $\ln x$ is defined for $x>0$, ensure that the solution satisfies this condition.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined for positive real numbers. The natural logarithm, denoted as ln(x), specifically uses the base 'e' (approximately 2.718). Understanding how to manipulate logarithmic expressions is crucial for solving logarithmic equations, as it involves properties such as the product, quotient, and power rules.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. This means that any argument of a logarithm must be greater than zero. When solving logarithmic equations, it is essential to check the solutions against the original equation to ensure they fall within this domain, as any extraneous solutions can lead to invalid results.

Solving Logarithmic Equations

To solve logarithmic equations, one typically isolates the logarithmic term and then exponentiates both sides to eliminate the logarithm. This process often involves rearranging the equation and applying properties of logarithms. After finding potential solutions, it is important to verify them by substituting back into the original equation to ensure they are valid.