Solve each equation. Give solutions in exact form. log₄ (x³ + 37) = 3

Verified step by step guidance

1. Recognize that the equation is a logarithmic equation of the form \(\log_4 (x^3 + 37) = 3\), where the base of the logarithm is 4.
2. Recall the definition of logarithm: \(\log_b A = C\) means \(b^C = A\). Apply this to rewrite the equation as an exponential equation: \$4^3 = x^3 + 37$.
3. Calculate \$4^3\( (but do not simplify the final number as per instructions), so the equation becomes \)64 = x^3 + 37$.
4. Isolate the term with \(x\) by subtracting 37 from both sides: \$64 - 37 = x^3$.
5. Solve for \(x\) by taking the cube root of both sides: \(x = \sqrt[3]{64 - 37}\). This is the exact form of the solution.

Key Concepts

Properties of Logarithms

Logarithms are the inverse operations of exponentiation. Understanding how to manipulate logarithmic expressions, such as converting between logarithmic and exponential forms, is essential for solving equations involving logs. For example, log_b(a) = c can be rewritten as b^c = a.

Solving Exponential Equations

Once a logarithmic equation is converted to its exponential form, solving for the variable often involves isolating the variable in an equation with exponents. This may require taking roots or applying inverse operations to simplify and find exact solutions.

Domain Restrictions in Logarithmic Functions

The argument of a logarithm must be positive, so when solving log equations, it is crucial to consider domain restrictions. For example, in log_4(x^3 + 37), the expression x^3 + 37 must be greater than zero to ensure the solution is valid.