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. log₄(3x+2)=3

Verified step by step guidance

1. Identify the given logarithmic equation: $\log 4_{}(3x+2)=3$.
2. Rewrite the logarithmic equation in its equivalent exponential form using the definition of logarithms: $4_{}^3=\; 3x\; +\; 2$. This means the base 4 raised to the power 3 equals the argument of the log.
3. Calculate the value of $4_{}^3$ symbolically (do not compute the final number), so the equation becomes $64\; =\; 3x\; +\; 2$ (since 4^3 = 64).
4. Solve the linear equation for $x$: subtract 2 from both sides to get $64\; -\; 2\; =\; 3x$, then divide both sides by 3 to isolate $x$.
5. Check the domain of the original logarithmic expression: ensure that the argument $3x\; +\; 2$ is greater than 0, so solve $3x\; +\; 2\; >\; 0$ to find the valid values of $x$. Confirm that your solution satisfies this domain restriction.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows converting the logarithmic equation into an exponential form to solve for the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function requires the argument inside the log to be positive. Identifying and applying this restriction ensures that any solution found is valid and does not produce undefined expressions.

Exact and Approximate Solutions

After solving the equation exactly, it is often necessary to provide a decimal approximation. Using a calculator to round the solution to two decimal places helps interpret and communicate the answer clearly.