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. log3(x+4)=−3

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined for positive real numbers. The equation log_b(a) = c means that b^c = a, where b is the base of the logarithm. Understanding how to manipulate and solve logarithmic equations is essential for finding the value of x in the given problem.

Domain of Logarithmic Expressions

The domain of a logarithmic expression is restricted to positive values. For the equation log3(x+4), the argument (x+4) must be greater than zero, which leads to the condition x > -4. Recognizing and applying these domain restrictions is crucial to ensure that any solutions found are valid within the context of the original logarithmic equation.

Approximation of Solutions

After solving logarithmic equations, it may be necessary to approximate solutions using a calculator, especially when the exact solution is not easily expressible. This involves evaluating the logarithmic expression numerically to obtain a decimal value, which should be rounded to the specified number of decimal places, such as two in this case. This step is important for practical applications where numerical solutions are required.