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. Apply the natural logarithm (ln) to both sides of the equation to utilize the logarithm property that allows the exponent to be brought down: $\backslash ln\backslash left(7^\{x+2\}\backslash right)\; =\; \backslash ln(410)$.
3. Use the logarithm power rule to rewrite the left side: $(x+2)\; \backslash ln(7)\; =\; \backslash ln(410)$.
4. Isolate the variable term by dividing both sides by $\backslash ln(7)$: $x\; +\; 2\; =\; \backslash frac\{\backslash ln(410)\}\{\backslash ln(7)\}$.
5. Finally, solve for $x$ by subtracting 2 from both sides: $x\; =\; \backslash frac\{\backslash ln(410)\}\{\backslash ln(7)\}\; -\; 2$. This expression represents the exact solution in terms of natural logarithms.

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 solve for the variable.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials. Using properties like log(a^b) = b·log(a), we can take the logarithm of both sides of an equation to bring the exponent down, making it easier to solve for the variable.

Using Calculators for Approximation

After expressing the solution in terms of logarithms, calculators are used to find decimal approximations. Understanding how to use natural (ln) or common (log) logarithm functions on a calculator is essential for obtaining accurate numerical answers.