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 exponential equation: $3e{e}^{5x}=1977$.
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 bring down the exponent: $\backslash ln\backslash left(e^\{5x\}\backslash right)\; =\; \backslash ln\backslash left(\backslash frac\{1977\}\{3\}\backslash right)$.
4. Use the logarithmic identity $\backslash ln\backslash left(e^\{a\}\backslash right)\; =\; a$ to simplify the left side: $5x\; =\; \backslash ln\backslash left(\backslash frac\{1977\}\{3\}\backslash right)$.
5. Solve for $x$ by dividing both sides by 5: $x\; =\; \backslash frac\{1\}\{5\}\; \backslash ln\backslash left(\backslash frac\{1977\}\{3\}\backslash right)$. This expression gives the exact solution in terms of natural logarithms.

Key Concepts

Exponential Equations

An exponential equation involves variables in the exponent, such as e^(5x). Solving these requires isolating the exponential expression and then applying logarithms to both sides to solve for the variable.

Natural Logarithms

The natural logarithm (ln) is the inverse function of the exponential function with base e. It is used to solve equations where the variable is in the exponent, allowing us to rewrite e^(5x) as 5x = ln(value).

Using a Calculator for Approximations

After expressing the solution in logarithmic form, a calculator is used to find decimal approximations. This step involves evaluating logarithms and performing arithmetic to get a numerical answer rounded to the desired decimal places.