91Ó°ÊÓ

To solve logarithmic equations, follow these basic steps to understand and isolate the logarithms.
For example, let's start with the equation \(3 \times \log_3 x = \log_3 27\).
The first goal is to isolate the logarithms on one side. We can use the properties of logarithms to simplify.

In this case, using the property of logarithms that states \(a \log_b c = \log_b (c^a)\), we simplify it to \(\log_3 (x^3) = \log_3 27\).
This allows us to set the arguments equal to each other because the bases and logarithms are identical:
\(x^3 = 27\).
Now, solving this equation involves taking the cube root of both sides.
This gives us the result \(x = 3\).
By following these steps, you can break down and solve most logarithmic equations efficiently.

Understanding the properties of logarithms is essential for simplifying and solving logarithmic equations.
Here are a few key properties you should know:

• Product Rule: \(\log_b (MN) = \log_b M + \log_b N\)
• Quotient Rule: \(\log_b (M/N) = \log_b M - \log_b N\)
• Power Rule: \(\log_b (M^a) = a \log_b M\)

In the exercise, we used the power rule: \(a \log_b c = \log_b (c^a)\).
By applying this rule, we changed the left side from \(3 \log_3 x\) to \(\log_3 (x^3)\).
Recognizing and using these properties can help you simplify complex logarithmic expressions and equations.

The cube root is the value that, when multiplied by itself three times (cubed), gives the original number.
For example, the cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\).
Cube roots are written as \(\sqrt[3]{27}\).

In our exercise, after setting the arguments of the logarithms equal, we get \(x^3 = 27\).
To solve for \(x\), we take the cube root of both sides:
\(x = \sqrt[3]{27} = 3\).
The cube root operation can be used for any real number, and being familiar with it is helpful in many algebraic contexts.
Remember, taking the cube root is essentially the same as raising the number to the power of \(1/3\).
For instance, \(27^{1/3} = 3\), because \(3^3 = 27\).