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 exponential equation: \(5^{(2x+3)} = 3^{(x-1)}\).
2. Take the natural logarithm (ln) of both sides to bring down the exponents: \(\ln\left(5^{(2x+3)}\right) = \ln\left(3^{(x-1)}\right)\).
3. Use the logarithm power rule to move the exponents in front: \((2x+3) \ln(5) = (x-1) \ln(3)\).
4. Distribute the logarithms: \(2x \ln(5) + 3 \ln(5) = x \ln(3) - \ln(3)\).
5. Collect all terms involving \(x\) on one side and constants on the other: \(2x \ln(5) - x \ln(3) = - \ln(3) - 3 \ln(5)\), then factor out \(x\) and solve for \(x\).

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 to isolate the variable. Understanding how to manipulate and equate exponential expressions is essential for finding solutions.

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 exponents down and solve for the variable. Both natural (ln) and common (log) logarithms can be used depending on convenience.

Using Calculators for Approximation

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This step involves evaluating logarithmic expressions and rounding the result to a specified precision, such as two decimal places, to provide a practical numerical answer.