In mathematics, logarithmic equations occur when the logarithm of an expression is set equal to another logarithm or to a constant. A specific type of logarithmic equation involves solving for the variable when the logarithms have the same base. One such equation is log subscript 3 baseline (x + 12) = log subscript 3 baseline (5 x), and this article aims to guide you through the steps involved in finding its solution.
The key to solving logarithmic equations is to use the properties of logarithms to simplify the equation and isolate the variable. We’ll explore the steps in detail in the following sections.
Understanding Logarithmic Equations
A logarithm is an exponent that indicates the power to which a base must be raised to produce a given number. For example, log subscript 3 baseline 27 = 3 because 3^3 = 27.
Properties of Logarithms
To solve logarithmic equations, it’s essential to understand the following properties of logarithms:
- log subscript b baseline (a) = c if and only if b^c = a.
- log subscript b baseline (bc) = c + log subscript b baseline (b).
- log subscript b baseline (a/c) = log subscript b baseline (a) – log subscript b baseline (c).
Solving log subscript 3 baseline (x + 12) = log subscript 3 baseline (5 x)
To solve log subscript 3 baseline (x + 12) = log subscript 3 baseline (5 x), we’ll use the properties of logarithms:
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Step 1: Rewrite the equation
Using the property log subscript b baseline (a) = c if and only if b^c = a, we can rewrite the equation as 3^(log subscript 3 baseline (x + 12)) = 3^(log subscript 3 baseline (5 x)).
Step 2: Equate the exponents
Since the bases are the same, we can equate the exponents, giving us log subscript 3 baseline (x + 12) = log subscript 3 baseline (5 x).
Step 3: Solve for x
Using the property log subscript b baseline (bc) = c + log subscript b baseline (b), we can rewrite the equation as log subscript 3 baseline (x + 12) = log subscript 3 baseline 5 + log subscript 3 baseline x.
Equating the coefficients of log subscript 3 baseline x, we get 1 = 1, which is always true.
Therefore, the solution to the equation is x = x, which means that all real numbers are solutions.
Variations of Logarithmic Equations
While we solved log subscript 3 baseline (x + 12) = log subscript 3 baseline (5 x), there are other variations of logarithmic equations that you may encounter:
Logarithmic Equations with Different Bases
If the bases of the logarithms are different, you can use the change of base formula to convert them to the same base before solving.
Logarithmic Equations with Exponents
If the variable is in the exponent of the logarithm, you can use logarithmic differentiation to solve the equation.
Logarithmic Equations with Absolute Values
If the variable or the argument of the logarithm is inside an absolute value, you may need to consider different cases based on the sign of the variable.
Tips for Solving Logarithmic Equations
Here are some tips to help you solve logarithmic equations:
- Understand the properties of logarithms and how to use them.
- Simplify the equation as much as possible before solving.
- Consider different cases based on the signs and values involved.
- Check your solutions by plugging them back into the original equation.
FAQs: Solving log subscript 3 baseline (x + 12) = log subscript 3 baseline (5 x)
Q: What is the base of the logarithms in the equation log subscript 3 baseline (x + 12) = log subscript 3 baseline (5 x)?
A: The base of the logarithms is 3.
Q: What property of logarithms is used to rewrite the equation as 3^(log subscript 3 baseline (x + 12)) = 3^(log subscript 3 baseline (5 x))?
A: The property used is log subscript b baseline (a) = c if and only if b^c = a.
Q: Why can we equate the exponents when the bases are the same?
A: When the bases are the same, the exponents represent the powers to which the base is raised to produce the same result.
Q: What is the solution to the equation log subscript 3 baseline (x + 12) = log subscript 3 baseline (5 x)?
A: The solution is x = x, which means that all real numbers are solutions.
Q: Can there be multiple solutions to a logarithmic equation?
A: Yes, logarithmic equations can have multiple solutions, especially when the variable is in the exponent or the argument of the logarithm.
Conclusion
Solving logarithmic equations can be challenging, but with a clear understanding of logarithmic properties and a step-by-step approach, you can find the solutions effectively. Logarithmic equations are important in various fields such as chemistry, physics, and engineering, and mastering their solution techniques will benefit you in solving complex problems.