Problem 8 - Precalculus

Question

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\log _{5}(25)=2\)

Step-by-Step Solution

Answer

The equation \(\log _{5}(25)=2\) can be rewritten as \(5^{2}=25\).

Step 1: Identify the Components

In the expression \(\log _{5}(25)=2\), identify the base \(b=5\), the argument \(c=25\), and the result \(a=2\).

Step 2: Apply the Property

We use the property of logarithms and exponentials: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\). Substitute the values from Step 1: here, \(b=5\), \(a=2\), and \(c=25\).

Step 3: Rewrite in Exponential Form

Using the identified components and property, rewrite the given logarithmic equation as an exponential equation: \(5^{2} = 25\).

Key Concepts

Exponential Functions

Exponential functions are fundamental in mathematics and describe situations where a constant base is raised to a variable exponent. This type of function has the form \(f(x) = b^x\), where \(b\) is the base and \(x\) is the exponent. The base \(b\) is a positive number, and the most common base is \(e\), known as the natural base, approximately equal to 2.718.

Exponential functions are characterized by rapid growth or decay. For example, they model phenomena such as population growth and radioactive decay. A simple example of an exponential function is \(2^x\), where the function value doubles as \(x\) increases by 1.

• Growth: If \(b > 1\), the function grows.
• Decay: If \(0 < b < 1\), the function decays.

Rember, understanding exponential functions involves being comfortable with the concept of a base, which directly links to the next topic: logarithmic equations.

Logarithmic Equations

Logarithmic equations provide the inverse operation to exponentiation. A logarithm answers the question: "To what exponent must a base be raised to produce a certain value?"Given an equation \(b^a = c\), its equivalent logarithmic form is \(\log_{b}(c) = a\). Here, \(\log\) represents the logarithm, \(b\) is the base, \(c\) is the argument, and \(a\) is the result or exponent.

Logarithms are used to solve equations involving exponential growth or decay when the exponent is unknown. They turn multiplication into addition, simplifying calculations and solving equations. For example:

• To solve \(2^x = 32\), use logarithms to find \(x = \log_{2}(32)\).
• Tools like properties of logarithms and changes of base can simplify and solve complex problems.

Logarithmic equations give invaluable insight into the exponential relationship between numbers, leading us to understand the connection between base, argument, and logarithms.

Base and Argument Identification

Identifying the base and argument in logarithmic and exponential expressions is crucial for understanding their transformation from one form to another.

In the expression \(\log_b(c) = a\), the base \(b\) is the number that undergoes repeated multiplication to result in \(c\), known as the argument. The solution \(a\) is the number of times the base is multiplied by itself to reach the argument.

• Base (\(b\)): In \(\log_{5}(25) = 2\), the base is 5.
• Argument (\(c\)): The number 25 is the argument.
• Exponent (\(a\)): The result or exponent is 2.

Once you identify these components, it becomes easier to switch between logarithmic and exponential forms. For instance, from logarithmic \(\log_b(c) = a\) to exponential \(b^a = c\), knowing each component's role helps streamline the conversion process. This understanding can also facilitate solving more complicated logarithmic or exponential equations.