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. 7^0.3x=813

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a variable appears in the exponent. To solve these equations, one typically uses properties of exponents and logarithms. For example, if you have an equation like a^x = b, you can take the logarithm of both sides to isolate x. This is essential for finding the value of the variable in terms of known quantities.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for exponents in equations. The common logarithm (base 10) and the natural logarithm (base e) are the most frequently used types. For instance, if you have an equation like 7^0.3x = 813, you can apply logarithms to both sides to transform the equation into a linear form, making it easier to solve for x.

Calculator Use for Approximations

Using a calculator to obtain decimal approximations is crucial in solving exponential equations, especially when the solutions involve logarithms. After isolating the variable, you can input the logarithmic expression into a calculator to find a numerical value. This step is important for providing a practical answer, as many real-world applications require solutions in decimal form, often rounded to a specified number of decimal places.