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: $\backslash ln(5^x)\; =\; \backslash ln(17)$.
3. Apply the logarithmic identity $\backslash ln(a^b)\; =\; b\; \backslash ln(a)$ to rewrite the left side: $x\; \backslash ln(5)\; =\; \backslash ln(17)$.
4. Isolate $x$ by dividing both sides of the equation by $\backslash ln(5)$: $x\; =\; \backslash frac\{\backslash ln(17)\}\{\backslash ln(5)\}$.
5. Use a calculator to find the decimal values of $\backslash ln(17)$ and $\backslash ln(5)$, then divide to get the decimal approximation of $x$ rounded 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. Understanding the properties of exponents is essential to manipulate and solve these equations.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentiation, allowing us to solve for variables in exponents. The natural logarithm (ln) uses base e, while the common logarithm (log) uses base 10. Applying logarithms to both sides of an equation helps isolate the exponent and solve for the variable.

Using a Calculator for Approximation

After expressing the solution in logarithmic form, a calculator is used to find a decimal approximation. This step involves evaluating logarithmic expressions and rounding the result to the desired decimal places, ensuring the solution is both exact in form and practical for real-world use.