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. 5^x=17

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 5^x = 17, the goal is to isolate x by applying logarithmic functions, which allows us to rewrite the equation in a more manageable form.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for exponents in equations. The logarithm of a number is the exponent to which a base must be raised to produce that number. In the context of the equation 5^x = 17, we can use either natural logarithms (ln) or common logarithms (log) to express x as x = log(17) / log(5) or x = ln(17) / ln(5).

Calculator Use for Approximations

Using a calculator to find decimal approximations of logarithmic values is essential for practical applications. After expressing the solution in logarithmic form, a calculator can provide a numerical value for x, which is often required in real-world scenarios. For instance, calculating log(17) / log(5) will yield a decimal approximation of the solution to the original exponential equation.