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

Verified step by step guidance

1. Identify the given logarithmic equation: \(\log_{2}(x + 25) = 4\).
2. Recall the definition of logarithm: \(\log_{b}(A) = C\) means \(b^{C} = A\). Apply this to rewrite the equation as \$2^{4} = x + 25$.
3. Calculate \$2^{4}\( (without final numeric evaluation here) and set up the equation \)x + 25 = 2^{4}$.
4. Solve for \(x\) by isolating it: \(x = 2^{4} - 25\).
5. Check the domain restriction for the logarithm: the argument \(x + 25\) must be greater than 0, so ensure \(x + 25 > 0\) and verify that your solution satisfies this condition.

Key Concepts

Properties of Logarithms

Understanding the basic properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows you to rewrite logarithmic equations in exponential form to solve for the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) requires that the argument x be positive. When solving equations, it is crucial to check that solutions do not make the argument of any logarithm zero or negative, as these are not valid.

Exact and Approximate Solutions

After finding the exact solution to a logarithmic equation, it is often necessary to provide a decimal approximation. Using a calculator to round the solution to a specified number of decimal places helps interpret and communicate the result clearly.