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. 10^x = 7000

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions in which a variable appears in the exponent. To solve these equations, one typically isolates the exponential term and may use logarithms to bring the variable down from the exponent. For example, in the equation 10^x = 7000, the goal is to find the value of x that satisfies this equality.

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. In the case of the equation 10^x = 7000, we can apply the common logarithm (base 10) to both sides, transforming the equation into x = log(7000).

Calculator Use for Approximations

Using a calculator to find decimal approximations is essential in solving exponential equations, especially when the solutions are not whole numbers. After applying logarithms, the resulting value may need to be calculated to a specific decimal place for practical applications. For instance, after finding x = log(7000), a calculator can provide the numerical value of x rounded to two decimal places.