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

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined as the power to which a base must be raised to obtain a given number. For example, in the equation log_b(a) = c, b^c = a. Understanding how to manipulate and solve logarithmic equations is crucial for finding the value of x in the given problem.

Domain of Logarithmic Expressions

The domain of a logarithmic expression is the set of all possible input values (x) for which the logarithm is defined. Specifically, the argument of the logarithm must be positive. In the equation log4(3x+2)=3, it is essential to ensure that 3x + 2 > 0 to find valid solutions, as any value outside this domain must be rejected.

Exact vs. Approximate Solutions

In solving logarithmic equations, an exact solution is typically expressed in terms of logarithms or algebraic expressions, while an approximate solution is a numerical value obtained through calculation. For instance, after solving log4(3x+2)=3 for x, one may need to use a calculator to find a decimal approximation, ensuring it is rounded to the specified number of decimal places.