What is x if 2^x = 3^{x-1}?
This article will discuss the mathematical equation 2^x = 3^{x-1} and how to solve for x. We will look at the equation, explain the steps to solve it, and provide examples to illustrate the process.
Equation 2^x = 3^{x-1} is an exponential equation
Isolate the variable x by dividing both sides of the equation by 2^x
Take the logarithm of both sides of the equation
Solve for x by adding xlog2 to both sides of the equation and dividing by log3
Solution to the equation is x = log2/log3, which is approximately 1.585
The equation 2^x = 3^{x-1} is an exponential equation. Exponential equations are equations that involve exponents, which are numbers that are multiplied by themselves a certain number of times. In this equation, the base of the exponent is 2 and the exponent is x. The other side of the equation has a base of 3 and an exponent of x-1.
To solve this equation, we must first isolate the variable x. To do this, we must divide both sides of the equation by 2^x. This will leave us with 3^{x-1}/2^x = 1. We can then take the logarithm of both sides of the equation. The logarithm of 3^{x-1}/2^x is (x-1)log3 - xlog2 = 0. We can then solve for x by adding xlog2 to both sides of the equation and dividing by log3. This will give us x = log2/log3.
To illustrate this process, let's look at an example. Suppose we have the equation 2^x = 3^{x-1}. We can isolate the variable x by dividing both sides of the equation by 2^x. This will leave us with 3^{x-1}/2^x = 1. We can then take the logarithm of both sides of the equation. The logarithm of 3^{x-1}/2^x is (x-1)log3 - xlog2 = 0. We can then solve for x by adding xlog2 to both sides of the equation and dividing by log3. This will give us x = log2/log3, which is approximately 1.585.
Good to know:
Exponential equation: An equation that involves exponents
Base: The number that is multiplied by itself a certain number of times
Exponent: The number of times the base is multiplied by itself
Logarithm: The inverse of an exponential equation
In conclusion, the equation 2^x = 3^{x-1} can be solved by isolating the variable x, taking the logarithm of both sides of the equation, and then solving for x. This process is illustrated in the example provided. The solution to the equation is x = log2/log3, which is approximately 1.585.
The information provided in this article is for educational purposes only and should not be used as a substitute for professional advice.
