In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2^(4x-2) = 64

Verified step by step guidance

1. Rewrite the equation in exponential form: \(2^{4x-2} = 64\). Recognize that 64 can be expressed as a power of 2, specifically \(64 = 2^6\). This gives \(2^{4x-2} = 2^6\).
2. Since the bases are the same, set the exponents equal to each other: \(4x - 2 = 6\).
3. Solve for \(x\) by isolating it. First, add 2 to both sides of the equation: \(4x = 8\).
4. Divide both sides of the equation by 4 to solve for \(x\): \(x = \frac{8}{4}\).
5. Simplify the fraction to find \(x\). If needed, verify the solution by substituting \(x\) back into the original equation.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions in which a variable appears in the exponent. To solve these equations, one often needs to rewrite them in a form that allows for easier manipulation, such as expressing both sides with the same base or using logarithms. Understanding the properties of exponents is crucial for simplifying and solving these types of equations.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in an exponential equation. The logarithm of a number is the exponent to which a base must be raised to produce that number. Familiarity with natural logarithms (base e) and common logarithms (base 10) is essential, especially when expressing solutions in logarithmic form.

Calculator Use for Approximations

Using a calculator to obtain decimal approximations is often necessary when dealing with logarithmic solutions or complex exponential equations. Understanding how to input logarithmic functions and interpret the results is vital for accurately finding and rounding solutions to the required decimal places, ensuring precision in mathematical communication.