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 by taking the natural logarithm (ln) of both sides of the equation: $\ln (5^{2x+3})=\ln (3^{x-1})$.
2. Use the power rule of logarithms to bring down the exponents: $(2x+3)\ln (5)=(x-1)\ln (3)$.
3. Distribute the logarithms: $2x\ln (5)+3\ln (5)=x\ln (3)-\ln (3)$.
4. Rearrange the equation to isolate terms involving $x$: $2x\ln (5)-x\ln (3)=-3\ln (5)-\ln (3)$.
5. Factor out $x$ from the left side: $x(2\ln (5)-\ln (3))=-3\ln (5)-\ln (3)$.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear in the exponent. To solve these equations, one typically uses properties of exponents and logarithms. For example, if you have an equation like a^x = b, you can take the logarithm of both sides to isolate x. Understanding how to manipulate these equations is crucial for finding solutions.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for exponents in equations. The natural logarithm (ln) and common logarithm (log) are the two most commonly used types. For instance, if a^x = b, then x can be expressed as x = log_a(b). Familiarity with logarithmic properties, such as the product, quotient, and power rules, is essential for solving exponential equations.

Calculator Use for Approximations

Using a calculator to obtain decimal approximations is a practical step after solving an equation symbolically. Most scientific calculators can compute logarithms and exponentials, which helps in finding numerical solutions. For example, after isolating x in an equation, you can input the logarithmic expression into the calculator to get a decimal value, rounding it to the desired precision, such as two decimal places.