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. 5^(2x+3)=3^(x−1)

Verified step by step guidance

1. Start with the given equation: $5^{2x+3}=3^{x-1}$.
2. Take the natural logarithm (ln) of both sides to utilize the property that ln(a^b) = b * ln(a): $\backslash ln\backslash left(5^\{2x+3\}\backslash right)\; =\; \backslash ln\backslash left(3^\{x-1\}\backslash right)$.
3. Apply the logarithm power rule to bring down the exponents: $(2x\; +\; 3)\; \backslash ln\; 5\; =\; (x\; -\; 1)\; \backslash ln\; 3$.
4. Distribute the logarithms and rewrite the equation as: $2x\; \backslash ln\; 5\; +\; 3\; \backslash ln\; 5\; =\; x\; \backslash ln\; 3\; -\; \backslash ln\; 3$.
5. Collect all terms involving $x$ on one side and constants on the other: $2x\; \backslash ln\; 5\; -\; x\; \backslash ln\; 3\; =\; -\; \backslash ln\; 3\; -\; 3\; \backslash ln\; 5$. Then factor out $x$: $x\; (2\; \backslash ln\; 5\; -\; \backslash ln\; 3)\; =\; -\; \backslash ln\; 3\; -\; 3\; \backslash ln\; 5$. Finally, solve for $x$ by dividing both sides by $(2\; \backslash ln\; 5\; -\; \backslash ln\; 3)$.

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent, such as 5^(2x+3) = 3^(x−1). Solving these requires rewriting or applying logarithms since the variable cannot be isolated using basic algebraic operations.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials and help solve equations where the variable is an exponent. Using properties like log(a^b) = b·log(a) allows us to bring down exponents and solve for the variable.

Using Calculators for Approximation

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This step involves evaluating logarithms and performing arithmetic to get a numerical answer, often rounded to a specified decimal place.