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. e^(1−5x)=793

Verified step by step guidance

1. Start with the given exponential equation: $e^{1\; -\; 5x}=\; 793$.
2. Apply the natural logarithm (ln) to both sides of the equation to utilize the property that ln(e^y) = y. This gives: $\backslash ln\backslash left(e^\{1\; -\; 5x\}\backslash right)\; =\; \backslash ln(793)$.
3. Simplify the left side using the logarithm property: $1\; -\; 5x\; =\; \backslash ln(793)$.
4. Isolate the variable term by subtracting 1 from both sides: $-5x\; =\; \backslash ln(793)\; -\; 1$.
5. Solve for $x$ by dividing both sides by -5: $x\; =\; \backslash frac\{\backslash ln(793)\; -\; 1\}\{-5\}$. This expression represents the exact solution in terms of natural logarithms.

Key Concepts

Exponential Equations

An exponential equation is one in which the variable appears in the exponent. Solving such equations often involves rewriting the equation to isolate the exponential expression and then applying logarithms to both sides to solve for the variable.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials. They allow us to solve equations where the variable is an exponent by converting the exponential form into a logarithmic form. Common logarithms (base 10) and natural logarithms (base e) are frequently used depending on the base of the exponential.

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 practical and usable answers.