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. log2(x+25)=4

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The equation log_b(a) = c means that b raised to the power of c equals a (b^c = a). Understanding this relationship is crucial for solving logarithmic equations, as it allows us to rewrite the logarithmic expression in exponential form to isolate the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. For the expression log_b(x), x must be greater than zero (x > 0) because logarithms of non-positive numbers are undefined. When solving logarithmic equations, it is essential to check that any solutions fall within this domain to ensure they are valid.

Approximation of Solutions

In many cases, logarithmic equations yield exact solutions that can be expressed in terms of logarithms. However, it is often useful to provide a decimal approximation for practical applications. This involves using a calculator to evaluate the logarithmic expression to a specified number of decimal places, which helps in understanding the solution in a more tangible way.