Problem 368 - Precalculus

Question

Use the definition of a logarithm along with the one-to-one property of logarithms to prove that \(b^{\log _{b} x}=x\)

Step-by-Step Solution

Answer

Using the definition, \(b^{\log_b x} = x\) holds true because \(\log_b x\) equals the exponent of \(b\) that gives \(x\).

Step 1: Understand Definitions

The definition of a logarithm states that if \(y = \log_b x\), then \(b^y = x\). This means that the logarithm \(\log_b x\) is the power to which the base \(b\) must be raised to get \(x\).

Step 2: Express Given Equation

To prove that \(b^{\log_b x} = x\), first express \(\log_b x\) as \(y\) such that \(y = \log_b x\). From this, we know that \(b^y = x\).

Step 3: Substitute Back into the Expression

Replace \(y\) in the expression \(b^y = x\) with \(\log_b x\) to get \(b^{\log_b x} = x\). Since \(\log_b x\) is defined as the exponent that yields \(x\) when \(b\) is raised to it, this shows that \(b^{\log_b x} = x\) is true.

Key Concepts

Logarithm Properties

Logarithm properties are key in understanding how to manipulate and evaluate logarithmic expressions.
These properties provide valuable tools for simplifying complex equations and solving mathematical problems efficiently.
Here are a few fundamental properties:

• **Product Property:** The logarithm of a product is equal to the sum of the logarithms:
  \( \log_b (xy) = \log_b x + \log_b y \).
• **Quotient Property:**The logarithm of a quotient is the difference of the logarithms:
  \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \).
• **Power Property:** The logarithm of a power indicates that the exponent can be brought in front of the logarithm:
  \( \log_b (x^a) = a \cdot \log_b x \).
• **Change of Base Formula:** It allows logarithms to be calculated with a different base:
  \( \log_b x = \frac{\log_k x}{\log_k b} \), for any positive base \(k\).

These properties allow us to rearrange and simplify logarithmic expressions easily.
Understanding them is crucial for proving more complex equations, such as the task given: proving \(b^{\log_b x} = x\).
Using the properties of logarithms, especially the power property, helps demystify this relationship by directly associating the logarithmic operation with its corresponding exponential form.

Exponentiation

Exponentiation is the process of raising a number to a power, which is an important concept in both algebra and calculus.
It involves a base number and an exponent, where the base is multiplied by itself as many times as the exponent indicates.
For example, \(b^n\) means that the base \(b\) is raised to the power \(n\) by multiplying \(b\) by itself \(n-1\) more times.

• **Base:** The number being multiplied.
• **Exponent:** Indicates how many times the base is used as a factor.
• **Exponentiation Example:** If \(b = 5\) and \(n = 3\), then \(b^n = 5^3 = 5 \times 5 \times 5 = 125\).

In the expression \(b^{\log_b x}\), something interesting happens.
The base \(b\) is raised to the power defined by the logarithm \(\log_b x\), which equals \(x\).
In essence, the process of exponentiation here "undoes" the effect of taking the logarithm, illustrating the inverse relationship these operations have with each other.
Exponentiation is crucial to understanding logarithms because it shows the steps to arrive at \(x\) from \(b^{\log_b x}\).
Recognizing this link helps clarify why logarithmic functions are designed the way they are.

One-to-One Property

The one-to-one property is an important characteristic for understanding and manipulating functions.
It ensures that each input corresponds to exactly one output, and each output is linked to only one input, making it bijective.
For logarithms, this property is particularly helpful in proving statements and solving equations.

• **Characteristic:** For a one-to-one function, \(f(a) = f(b)\) implies \(a = b\).
• **Application in Logarithms:** If \(\log_b (m) = \log_b (n)\), it follows that \(m = n\) because the base \(b\) raised to these outputs must yield the same result.
• **In Exponentiation:** Since \(b^{\log_b x} = x\), this shows that the function is undoing the logging process precisely back to the original number \(x\).

The one-to-one nature of logarithmic functions simplifies proving correctness and verifying steps.
When dealing with equations like \(b^{\log_b x} = x\), it reaffirms the idea that these processes can reverse one another perfectly without losing information.
Understanding this property is vital as it forms the basis for many actions in algebra involving inverse functions and operations.
Mastering the one-to-one property offers a clearer insight into how logarithms and exponentiation coexist and operate within mathematical frameworks.