Solve each exponential equation. Express the solution set in...

Solve each exponential equation. 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.
$$5^{2 x+3}=3^{x-1}$$

Key Concepts

Exponential Equations

Exponential equations are equations where the variable appears in the exponent. Solving such equations typically requires techniques to 'bring down' the exponent so that the variable can be isolated and solved using algebraic methods.

Logarithms

Logarithms are the inverses of exponential functions and serve as crucial tools in solving equations with variables in exponents. They allow one to convert multiplicative relationships into additive ones, making equations easier to handle.

Properties of Logarithms

The properties of logarithms, especially the power rule (stating that log(a^b) = b*log(a)), are essential for rewriting exponential expressions in a more manageable form. These properties facilitate isolating the variable within the equation.

Using Logarithms to Solve Equations

By taking the logarithm of both sides of an exponential equation, one can leverage logarithmic properties to bring down exponents, thereby transforming the equation into a linear form. This linear form can then be manipulated using standard algebraic techniques to solve for the variable.

Decimal Approximations of Logarithmic Solutions

Once a solution is obtained in terms of logarithms, a calculator is used to evaluate the logarithmic expressions numerically. This step provides a decimal approximation of the solution, which is often necessary for practical applications or verifying the answer.

Transcript

Please give Ace some feedback