Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 7^(x+2)=410

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions in which a variable appears in the exponent. To solve these equations, one typically uses properties of exponents and logarithms. For example, in the equation 7^(x+2) = 410, the goal is to isolate the variable x by applying logarithmic functions.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in an exponential equation. The common logarithm (base 10) and the natural logarithm (base e) are frequently used. In the context of the given equation, taking the logarithm of both sides helps to bring down the exponent, facilitating the solution process.

Calculator Use for Approximations

Using a calculator to find decimal approximations is essential for providing practical solutions to exponential equations. After isolating the variable, one can compute the logarithmic values to obtain a numerical solution. In this case, after solving for x, a calculator will yield a decimal approximation, which is often rounded to a specified number of decimal places for clarity.