Solve each equation. Give solutions in exact form. See Examples 5–9. log_4 (x^3 + 37) = 3

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Start by understanding the equation: \( \log_4 (x^3 + 37) = 3 \). This is a logarithmic equation where the base is 4.

To eliminate the logarithm, rewrite the equation in its exponential form: \( x^3 + 37 = 4^3 \).

Calculate \( 4^3 \) to find the value on the right side of the equation.

Subtract 37 from both sides to isolate \( x^3 \): \( x^3 = 4^3 - 37 \).

Finally, solve for \( x \) by taking the cube root of both sides: \( x = \sqrt[3]{4^3 - 37} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Properties of Logarithms

Logarithms have several properties that simplify calculations, such as the product, quotient, and power rules. For example, log_b(mn) = log_b(m) + log_b(n) and log_b(m/n) = log_b(m) - log_b(n). These properties are essential for manipulating logarithmic equations to isolate the variable.

Solving Exponential Equations

To solve an equation involving logarithms, we often convert it into an exponential form. In the given equation, log_4(x^3 + 37) = 3 can be rewritten as x^3 + 37 = 4^3. This transformation allows us to solve for x by isolating it and simplifying the resulting polynomial equation.

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