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^(5x−3) - 2 =10,476

Verified step by step guidance

1. Start by isolating the exponential expression on one side of the equation. Add 2 to both sides to get $e^{5x-3}=\; 10478$.
2. Apply the natural logarithm (ln) to both sides of the equation to undo the exponential function, using the property $\backslash ln(e^y)\; =\; y$. This gives $\backslash ln\backslash left(e^\{5x\; -\; 3\}\backslash right)\; =\; \backslash ln(10478)$.
3. Simplify the left side using the logarithm property to get $5x\; -\; 3\; =\; \backslash ln(10478)$.
4. Solve for $x$ by first adding 3 to both sides: $5x\; =\; \backslash ln(10478)\; +\; 3$, then divide both sides by 5: $x\; =\; \backslash frac\{\backslash ln(10478)\; +\; 3\}\{5\}$.
5. To find the decimal approximation, use a calculator to evaluate $\backslash ln(10478)$, then perform the arithmetic operations as indicated, rounding the final answer to two decimal places.

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent, such as e^(5x−3). Solving these requires isolating the exponential expression and then applying logarithms to both sides to solve for the variable.

Natural Logarithms

Natural logarithms (ln) are the inverse of the exponential function with base e. They are used to solve equations where the variable is in the exponent of e, allowing us to rewrite e^y = a as y = ln(a).

Using Calculators for Approximations

After expressing the solution in logarithmic form, calculators help find decimal approximations. This step involves evaluating logarithms and rounding the result to the desired decimal places, ensuring practical and precise answers.