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. ln√x+3=1

Verified step by step guidance

1. Start by isolating the logarithmic expression: $\ln (\sqrt{x}+3)=1$.
2. To eliminate the natural logarithm, exponentiate both sides using base $e$: $e^{\ln (\sqrt{x}+3)}=e^1$.
3. This simplifies to: $\sqrt{x}+3=e$.
4. Subtract 3 from both sides to isolate $\sqrt{x}$: $\sqrt{x}=e-3$.
5. Square both sides to solve for $x$: $x=(e-3{)}^2$.

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', specifically uses the base 'e' (approximately 2.718). Understanding how to manipulate logarithmic expressions, including properties like the product, quotient, and power rules, is essential for solving logarithmic equations.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive values. For the equation ln(√x + 3) = 1, it is crucial to ensure that the argument of the logarithm, √x + 3, is greater than zero. This means that x must be greater than or equal to -3, but since √x must also be non-negative, x must be at least 0. Identifying the domain helps in rejecting any extraneous solutions.

Solving Logarithmic Equations

To solve logarithmic equations, one typically converts the logarithmic form into its exponential form. For example, ln(a) = b can be rewritten as a = e^b. After isolating the variable, it is important to check the solutions against the domain restrictions to ensure they are valid. Additionally, using a calculator for decimal approximations may be necessary for practical applications.