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(x+5)=3

Verified step by step guidance

1. Identify the given logarithmic equation: \(\log_{4}(x+5) = 3\).
2. Recall the definition of logarithm: \(\log_{a}(b) = c\) means \(a^{c} = b\). Apply this to rewrite the equation as \$4^{3} = x + 5$.
3. Calculate the value of \$4^{3}\( (but do not finalize the numeric value here), so the equation becomes \)x + 5 = 4^{3}$.
4. Isolate \(x\) by subtracting 5 from both sides: \(x = 4^{3} - 5\).
5. Check the domain restriction: since the argument of the logarithm \(x + 5\) must be greater than 0, ensure that \(x + 5 > 0\), which means \(x > -5\). Confirm that your solution satisfies this domain condition.

Key Concepts

Definition and Properties of Logarithms

A logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding this definition allows you to rewrite logarithmic equations in exponential form, which is essential for solving equations like log_4(x+5) = 3.

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+5 > 0 in the given problem, so x must be greater than -5. Checking the domain ensures that any solution found is valid within the original equation.

Converting Logarithmic Equations to Exponential Form

To solve log_b(f(x)) = c, rewrite it as f(x) = b^c. For log_4(x+5) = 3, this becomes x+5 = 4^3. This conversion simplifies the problem to solving an algebraic equation, making it easier to find the exact value of x.