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. Isolate the logarithmic term by subtracting 6 from both sides: $6+2\ln x=5\backslash implies2\ln x=5-6=-1$.
2. Divide both sides of the equation by 2 to solve for $\ln x$: $\ln x=\; \backslash frac\{-1\}\{2\}\; =\; -\backslash frac\{1\}\{2\}$.
3. Rewrite the equation in exponential form to solve for $x$: $x=\; e^\{-\backslash frac\{1\}\{2\}\}$, since $\backslash ln\; x\; =\; y\; \backslash iff\; x\; =\; e^y$.
4. Check the domain of the original logarithmic expression: since $\backslash ln\; x$ is defined only for $x\; >\; 0$, verify that the solution satisfies this condition.
5. If needed, use a calculator to approximate $e^\{-\backslash frac\{1\}\{2\}\}$ to two decimal places for a decimal approximation of the solution.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the ability to isolate the logarithmic term and use inverse operations, is essential. For example, knowing that ln(x) can be isolated and then exponentiated to solve for x helps in solving equations like 6 + 2 ln x = 5.

Domain of Logarithmic Functions

The domain of the natural logarithm function ln(x) is x > 0. When solving logarithmic equations, it is crucial to check that the solutions fall within this domain to avoid extraneous or invalid answers.

Solving Linear Equations

Before isolating the logarithmic term, you often need to solve a linear equation involving constants and logarithmic expressions. For example, subtracting 6 from both sides and dividing by 2 simplifies the equation, making it easier to solve for ln(x).