Solving Exponential Equations: 3^x = 5^x - 2
In this article, we will explore the solution to the exponential equation 3^x = 5^x - 2. Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and their properties.
What is an Exponential Equation?
An exponential equation is an equation in which the variable (in this case, x) is in the exponent. In other words, the equation involves a base number raised to a power that is a function of the variable. Exponential equations can be expressed in the form:
a^x = b^x + c
where a, b, and c are constants, and x is the variable.
Solving the Equation 3^x = 5^x - 2
To solve the equation 3^x = 5^x - 2, we can start by noticing that both sides of the equation involve exponential functions with different bases (3 and 5). This suggests that we can use the property of exponential functions that states:
a^x = b^x ⇔ x = loga(b)
where loga is the logarithm with base a.
Taking Logarithms
Taking the logarithm with base 3 on both sides of the equation, we get:
x = log3(5^x - 2)
This equation involves a composition of logarithmic and exponential functions. To simplify it, we can use the property of logarithms that states:
loga(b^x) = x * loga(b)
Applying this property to the right-hand side of the equation, we get:
x = log3(5^x) - log3(2)
x = x * log3(5) - log3(2)
Simplifying the Equation
Simplifying the equation, we get:
x = x * log3(5) - log3(2)
x - x * log3(5) = -log3(2)
x(1 - log3(5)) = -log3(2)
x = -log3(2) / (1 - log3(5))
Approximating the Solution
Using a calculator to approximate the values of log3(2) and log3(5), we get:
x ≈ -0.431 / (-0.221)
x ≈ 1.95
Therefore, the approximate solution to the equation 3^x = 5^x - 2 is x ≈ 1.95.
Conclusion
In this article, we have demonstrated how to solve the exponential equation 3^x = 5^x - 2 using logarithmic and exponential properties. The solution involves taking logarithms, simplifying the equation, and approximating the value of x. This exercise highlights the importance of understanding exponential functions and their properties in solving equations.
