Problem 78 - College Algebra

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

By using the definition of logarithms, we see that \(b^{\log_b x} = x\).

Step 1: Understanding the Problem

The exercise requires us to use the definition of the logarithm and its properties to show that if you raise a base to a logarithm with the same base, you'll end up with what's inside the logarithm, i.e., show that \(b^{\log_b x} = x\).

Step 2: Recall the Definition of a Logarithm

The definition of a logarithm states that \(\log_b x = y\) if and only if \(b^y = x\). This means that the logarithm function asks the question: "To what power must we raise \(b\) to get \(x\)?"

Step 3: Applying the One-to-One Property of Logarithms

The one-to-one property of logarithms states that if \(\log_b M = \log_b N\), then \(M = N\). This implies that logarithmic expressions with the same base and equal values have the same argument.

Step 4: Apply the Definition to Simplify the Expression

Given the expression \(b^{\log_b x}\), we want to consider \(y = \log_b x\), which means \(b^y = x\). Substitute \(y\) in the expression: \(b^y = b^{\log_b x} = x\).

Step 5: Conclude the Proof

Using the given properties and definitions, we've shown that \(b^{\log_b x} = x\). The properties of exponents and logarithms guarantee that the original power reduces to the argument \(x\), validating that both sides are equal.

Key Concepts

Definition of a Logarithm

Logarithms might seem intimidating at first, but they're actually quite straightforward when you think about them in terms of exponentiation. The definition of a logarithm is: if you have \(\log_b x = y\), it means that \(b^y = x\). Essentially, a logarithm tells you the power to which a base number \(b\) must be raised to produce another number, \(x\).
\(\log_b x\) can be thought of as the inverse of \(b^y = x\). When you see \(\log_b x\), you're essentially asking, "What power do I need to raise \(b\) to in order to get \(x\)?" For example, if \(b = 2\), \(x = 8\), then \(\log_2 8 = 3\) because \(2^3 = 8\).

Logarithms are a crucial tool in mathematics and science because they help simplify complex multiplicative relationships into manageable additive ones. This makes calculations more intuitive and easier to perform by hand.

One-to-One Property of Logarithms

The one-to-one property is an important characteristic of logarithmic functions. This property states that for a function \(\log_b M = \log_b N\), it must follow that \(M = N\).
With this property, we understand that logarithmic functions with the same base are perfectly consistent; they produce a unique output for a given input. There's no mix-up; if the outputs (logs) are equal, the inputs must have been the same too.

This property is vital when it comes to solving equations involving logarithms. If you encounter an equation like \(\log_3 (x+1) = \log_3 4\), you can confidently conclude that \(x + 1 = 4\) due to the one-to-one property.
When applying this to exponential-logarithmic expressions, it is crucial to establish that only one value of \(M\) can satisfy a particular log equation without ambiguity.

Proof by Applying Properties

Proving a statement using logarithmic and exponential properties involves a certain elegance due to the precision and clarity these properties offer.
Let's demonstrate this with a common identity: proving \(b^{\log_b x} = x\).
First, recall that by the definition of logarithms, for \(\log_b x = y\), it follows that \(b^y = x\). When faced with \(b^{\log_b x}\), substitute \(y = \log_b x\). Thus, \(b^{\log_b x} = b^y = x\).
This demonstrates how using the definition of the logarithm alongside the property of exponents proves the identity. Equally important is the one-to-one property of logarithms, ensuring that if \(\log_b M = \log_b N\), then \(M = N\), bringing consistency to this proof.
The process comes down to substituting correctly and understanding the intimate relationship between exponentiation and logarithms. These are transformative principles in mathematics, helping unravel complex problems into understandable solutions.