91Ó°ÊÓ

Logarithmic identities are useful tools for solving complex equations, especially when they involve exponential functions.
One of the most important identities you need to remember is: \(a^ {\log _a (b)} = b\).
This identity tells us that if you have a logarithm with a base \(a\) and you raise it to a power of \(a\log a (b)\), it simplifies to \(b\).
In our example, the base is 5 and the expression inside the logarithm is 8. So, \(5^{\log _{5} 8}= 8\).
This identity simplifies the equation and allows us to solve for the variable more directly.

Solving for variables involves isolating the variable on one side of the equation.
Once we've simplified the equation using logarithmic identities, our equation becomes much simpler:
\(8 = 2x\).
Now, to solve for \(x\), we need to isolate it on one side of the equation. This usually involves inverse operations.
In this case, we divide both sides by 2 to get: \(x = \frac{8}{2}\).
Simplifying the right side, we find \(x = 4\).

Exponential functions are mathematical functions in the form \(a^x\), where \(a\) is a constant base and \(x\) is the exponent.
These functions are crucial in various fields such as science and finance.
Let's revisit our original problem: \(5^{\log _{5} 8} = 2x\).
The exponential function here involves a base of 5 raised to a power of a logarithmic expression. By using logarithmic identities, we simplified this to 8.
The simplified equation then allowed us to easily solve for \(x\) by isolating it through division.
Understanding exponential functions and how they behave with logarithms helps us break down and solve these equations more effectively.
In this case, recognizing the relationship allowed us to simplify \(5^{\log _{5} 8}\) to just 8, making it straightforward to solve for \(x\).