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. 7^(x+2)=410

Verified step by step guidance

1. Start with the given exponential equation: \$7^{(x+2)} = 410$.
2. To solve for \(x\), take the natural logarithm (or common logarithm) of both sides to utilize the property that \(\ln(a^b) = b \ln(a)\): \(\ln\left(7^{(x+2)}\right) = \ln(410)\).
3. Apply the logarithm power rule to bring down the exponent: \((x+2) \ln(7) = \ln(410)\).
4. Isolate the term with \(x\) by dividing both sides by \(\ln(7)\): \(x + 2 = \frac{\ln(410)}{\ln(7)}\).
5. Finally, solve for \(x\) by subtracting 2 from both sides: \(x = \frac{\ln(410)}{\ln(7)} - 2\).

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 (Natural and Common)

Logarithms are the inverse operations of exponentiation. The natural logarithm (ln) uses base e, while the common logarithm (log) uses base 10. Applying logarithms allows us to solve for variables in exponents by converting the equation into a linear form.

Using a Calculator for Decimal Approximations

After expressing the solution in logarithmic form, a calculator is used to find decimal approximations. This step involves evaluating logarithmic expressions and rounding the result to the desired decimal places, ensuring practical and understandable answers.