Problem 21 - Precalculus with Calculus Previews

Question

In Problems \(21-40,\) solve the given logarithmic equation. $$ \log _{3} 5 x=\log _{3} 160 $$

Step-by-Step Solution

Answer

The solution is \( x = 32 \).

Step 1: Understand the Logarithmic Equation

We start with the equation \( \log_{3} 5x = \log_{3} 160 \). This is a standard logarithmic equation where both sides have the same base, which is 3.

Step 2: Leverage the Property of Logarithms

The property of logarithms states that if \( \log_{b} A = \log_{b} B \), then \( A = B \). Since both sides of our equation are logarithms with the same base (3), we can set their arguments equal: \( 5x = 160 \).

Step 3: Solve for the Variable

To solve for \( x \), we divide both sides of the equation \( 5x = 160 \) by 5:\[x = \frac{160}{5} = 32\]

Key Concepts

Properties of Logarithms

Logarithms have several important properties that are incredibly useful when solving equations. One such property is called the "Equality Property of Logarithms." This property states that if two logarithms with the same base are equal, then their arguments must also be equal.
In mathematical terms, if \( \log_{b}(A) = \log_{b}(B) \), then \( A = B \). This is because a logarithm is essentially the inverse of an exponent. So, when two logarithms with the same base are equal, it implies that the numbers they represent are also equal.
Additionally, other properties like the Product Rule, Quotient Rule, and Power Rule allow us to manipulate and simplify logarithmic expressions, making equations easier to solve. Understanding and applying these properties are fundamental in tackling more complex logarithmic equations.

Solving Logarithmic Equations

To solve logarithmic equations, you typically apply the properties of logarithms to simplify the equation or to combine terms. In our example, the equation is \( \log_{3} 5x = \log_{3} 160 \). Both sides of the equation have the same logarithmic base. This allows us to use the Equality Property of Logarithms to simplify the problem.

• First, identify if the logarithms have the same base, which will allow you to set their arguments equal, as the property describes.
• Next, once the arguments are set equal, it turns the logarithmic equation into a simple algebraic equation, such as \( 5x = 160 \).
• Finally, solve this algebraic equation for the unknown variable, which in this case involves basic arithmetic.

Mastering these strategies can streamline your problem-solving process in a variety of logarithmic contexts.

Base of Logarithms

The base of a logarithm is a key element that defines the characteristics of a logarithmic function. In our equation, the base is \( 3 \). This denotes that the logarithm \( \log_{3}(x) \) is asking "To what power must 3 be raised, to yield \( x \)?". The base of the logarithm is always a positive number different from 1.
Understanding the base is crucial because it affects how you interpret the logarithmic expression.

• Changing the base modifies the entire logarithmic function.
• It's important to note that logarithms can have any positive number as a base, though common bases are 10 (common logarithm) and \( e \) (natural logarithm).

In exercises like our example, the base being consistent across the equation allows us to directly equate the arguments, simplifying the process.