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: $\log {\mathrm{b}}_{}\left(a\right)=\; c$ means $b{^}^c=a$. Apply this to rewrite the equation in exponential form: $2_{}{^}^4=x+25$.
3. Calculate the value of $2_{}{^}^4$ (which is 2 raised to the 4th power) to simplify the equation to $16=x+25$.
4. Solve for $x$ by isolating it on one side: subtract 25 from both sides to get $x=\; 16\; -\; 25$.
5. Check the domain of the original logarithmic expression: since the argument of the logarithm $x+\; 25$ must be greater than 0, ensure that the solution satisfies $x+\; 25\; >\; 0$. If it does, the solution is valid; if not, reject it.

Key Concepts

Properties of Logarithms

Understanding the definition and properties of logarithms is essential. A logarithm log_b(a) answers the question: to what power must the base b be raised to get a? For example, log2(x+25) = 4 means 2 raised to 4 equals x+25. This allows converting logarithmic equations into exponential form for easier solving.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(f(x)) requires that the argument f(x) be positive. This means x+25 > 0, so x must be greater than -25. Checking the domain ensures that any solution found is valid and does not make the logarithm undefined.

Solving Exponential Equations

After rewriting the logarithmic equation in exponential form, solving for x involves basic algebraic manipulation. For log2(x+25) = 4, rewrite as x+25 = 2^4, then solve for x by subtracting 25. This step finds the exact solution before verifying domain constraints.