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

Verified step by step guidance

1. Start with the given exponential equation: \$5^x = 17$.
2. To solve for \(x\), take the natural logarithm (ln) of both sides to utilize the logarithm property that allows exponents to be brought down: \(\ln(5^x) = \ln(17)\).
3. Apply the logarithm power rule: \(x \cdot \ln(5) = \ln(17)\).
4. Isolate \(x\) by dividing both sides of the equation by \(\ln(5)\): \(x = \frac{\ln(17)}{\ln(5)}\).
5. Use a calculator to find the decimal values of \(\ln(17)\) and \(\ln(5)\), then divide to get the decimal approximation of \(x\), rounding to two decimal places.

Key Concepts

Exponential Equations

An exponential equation is one in which the variable appears in the exponent, such as 5^x = 17. Solving these equations often requires rewriting or applying logarithms to isolate the variable and find its value.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials, allowing us to solve for variables in exponents. Using properties like log(a^x) = x log(a), we can rewrite exponential equations in logarithmic form to isolate the variable.

Using Calculators for Logarithmic Approximations

After expressing the solution in logarithmic form, calculators help find decimal approximations. Both natural logarithms (ln) and common logarithms (log) can be used, and the result is rounded to the desired decimal places for practical answers.