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

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a variable appears in the exponent. To solve these equations, one often needs to manipulate the equation to isolate the variable. This can involve rewriting the equation in a form that allows for the application of logarithms, which are the inverse operations of exponentiation.

Logarithms

Logarithms are the inverse functions of exponentiation, allowing us to solve for the exponent in an exponential equation. For example, if we have an equation of the form a^x = b, we can take the logarithm of both sides to find x = log_a(b). Understanding how to use natural logarithms (ln) and common logarithms (log) is essential for solving exponential equations.

Calculator Use for Approximations

Using a calculator to obtain decimal approximations is crucial when dealing with logarithmic solutions that do not yield simple fractions or integers. Most scientific calculators can compute logarithms and exponentials, allowing students to find numerical solutions to equations. It is important to round these approximations correctly, often to two decimal places, as specified in many problems.