91Ó°ÊÓ

Logarithmic equations are equations that involve logarithms with an unknown variable inside them. Understanding logarithmic properties is crucial to solving these types of equations. Such equations typically have the form \(\text{log}_b(a) = c\), where \(b\) is the base, \(a\) is the argument, and \(c\) is the result.
To solve these equations, consider the following steps:
1. **Identify the logarithmic equation.** This is apparent when you see \(\text{log}\) in the equation.
2. **Rewrite the equation in a different form** if necessary, often converting from logarithmic form to exponential form.
3. **Solve for the variable** by isolating it using algebraic techniques such as addition, subtraction, multiplication, and division.

To solve a logarithmic equation more easily, we often convert it to an exponential form. The phrase 'logarithm to exponential form' refers to using the property: \(\text{log}_b(a) = c\) is equivalent to \(b^c = a\). This conversion removes the logarithm, simplifying the equation.
For example, in the given exercise: \(\text{log}_5(12x-8) = 3\). Using the property, we rewrite it as:
\[5^3 = 12x - 8\]
Here,

• 5 is the base
• 3 is the exponent
• \(12x-8\) is the expression inside the logarithm

Now, the equation is easier to work with: \(125 = 12x - 8\).

After converting the logarithmic equation to exponential form, our goal is to isolate the variable. In the example, once we have \(125 = 12x - 8\), we need to solve for \(x\) by isolating it.
First, eliminate or move any constants away from the term containing \(x\).

• Add 8 to both sides: \[125 + 8 = 12x\]
• This simplifies to: \[133 = 12x\]

Next, perform the division to isolate \(x\):

• Divide both sides by 12: \[x = \frac{133}{12}\]

By simplifying this fraction, we have found the exact solution for \(x\). This approach is systematic and ensures that we methodically solve for the unknown variable.