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 with the given equation: \(e^{(5x - 3)} - 2 = 10476\).
2. Isolate the exponential term by adding 2 to both sides: \(e^{(5x - 3)} = 10476 + 2\).
3. Simplify the right side: \(e^{(5x - 3)} = 10478\).
4. Take the natural logarithm (ln) of both sides to undo the exponential: \(\ln\left(e^{(5x - 3)}\right) = \ln(10478)\).
5. Use the logarithm power rule to bring down the exponent: \((5x - 3) = \ln(10478)\), then solve for \(x\) by isolating it: \$5x = \ln(10478) + 3\(, and finally \)x = \frac{\ln(10478) + 3}{5}$.

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 and Common Logarithms

Natural logarithms (ln) use base e, while common logarithms (log) use base 10. They are inverse functions of exponential functions and are used to solve equations where the variable is an exponent.

Using a Calculator for Approximation

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, such as two decimals.