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. 3e^5x=1977

Verified step by step guidance

1. Start with the given equation: \$3e^{5x} = 1977$.
2. Isolate the exponential term by dividing both sides of the equation by 3: \(e^{5x} = \frac{1977}{3}\).
3. Apply the natural logarithm (ln) to both sides to undo the exponential function: \(\ln\left(e^{5x}\right) = \ln\left(\frac{1977}{3}\right)\).
4. Use the logarithmic identity \(\ln\left(e^{a}\right) = a\) to simplify the left side: \(5x = \ln\left(\frac{1977}{3}\right)\).
5. Solve for \(x\) by dividing both sides by 5: \(x = \frac{1}{5} \ln\left(\frac{1977}{3}\right)\).

Key Concepts

Exponential Equations

An exponential equation is one in which the variable appears in the exponent. Solving such equations often involves isolating 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, using properties like log(a^b) = b log(a).

Using Calculators for Approximations

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This involves evaluating natural (ln) or common (log) logarithms and rounding the result to the desired decimal places.