91Ó°ÊÓ

Understanding the properties of logarithms is crucial to solving logarithmic equations. Logarithms have several useful properties that can simplify complex problems.
One important property is the power rule, which states that for any positive number \(a\) and \(x\), \(a \, \log x = \log x^a\).
This rule allows us to move a multiplier in front of a log inside as an exponent. For example, \(3 \, \log x = \log x^3\). This is helpful for simplifying equations where a log is multiplied by a constant.
Using this property helps transform the given equation, \(3 \, \log x = \log 125\), into a more manageable form: \(\log x^3 = \log 125\). This simplification is key to making the equation easier to solve.

Equating arguments is a method used in logarithmic equations to simplify and solve for the variable. When both sides of an equation have logarithms with the same base, we can set their arguments equal.
That means if \(\log a = \log b\), then \(a = b\). This is because the logarithm function is one-to-one.
Applying this to our problem, after using the properties of logarithms, we have \(\log x^3 = \log 125\).
By equating the arguments, we get \(x^3 = 125\). This step is critical because it removes the logarithms and leaves us with a simple algebraic equation to solve.

A cube root is the value that, when multiplied by itself three times, gives the original number.
For instance, the cube root of 8 is 2 because \(2^3 = 8\). To solve \(x^3 = 125\), we need to find a number that when cubed equals 125.
Taking the cube root of both sides, we write this as \(x = \sqrt[3]{125}\).
To simplify \(\sqrt[3]{125}\), recognize that 125 can be expressed as \(5^3\).
Thus, \(\sqrt[3]{5^3} = 5\).
Therefore, the solution to the equation \(3 \, \log x = \log 125\) is \(x = 5\). Understanding cube roots helps us in solving polynomial equations involving powers and roots.