Problem 78 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Using the definition and one-to-one property of logarithms, we prove that \(b^{\log_b x} = x\).

Step 1: Understanding the Problem

We need to show that the expression \(b^{\log_b x} = x\) is true. This involves using the definition of logarithms and leveraging their properties. The base of the logarithm \(b\) is the same as the base of the exponent \(b\) in the expression.

Step 2: Using the Definition of Logarithm

The definition of the logarithm \(\log_b x = y\) tells us that it is equivalent to saying \(b^y = x\). In this problem, set \(y = \log_b x\). This implies that \(b^y = b^{\log_b x} = x\).

Step 3: Applying the One-to-One Property

The one-to-one property of logarithms states that if \(\log_b a = \log_b c\), then \(a = c\). Since we have \(b^{\log_b x} = x\) already from the definition of the logarithm, we can conclude that \(b^{\log_b x} = x\) is true, confirming the identity.

Key Concepts

Definition of Logarithm

Logarithms are mathematical functions that help us solve equations where the unknown is an exponent. The logarithm answers the question: "To what exponent must the base be raised, to produce a given number?" In terms of notation, you can express this relationship as \( \log_b x = y \\) which means the base \(b\) raised to the power \(y\) equals \(x\), or \(b^y = x\).

• The base \(b\) must be a positive number and not equal to 1.
• Logarithms convert multiplicative relationships into additive ones, facilitating easier computation.

In the context of the exercise, knowing the definition of a logarithm is key to understanding why \(b^{\log_b x} = x\). By setting \(y = \log_b x\), we are inherently stating that \(b^y = x\), thus confirming the expression with the definition alone.

One-to-One Property of Logarithms

The one-to-one property of logarithms is crucial for understanding how only one output corresponds to each input of a logarithmic function. This property is important because it means \( \log_b a = \log_b c \) implies \( a = c \).

• This makes logarithmic functions invertible; every output is paired with exactly one input.
• It ensures the consistency and reliability needed in solving equations involving logarithms.

For our exercise, this property reinforces that \(b^{\log_b x} = x\) is valid. After using the definition of the logarithm to establish that \(b^y = x\), we can assert this identity is correct through the one-to-one property, affirming that no other value but \(x\) satisfies the equation

Exponential Function Properties

Exponential functions are characterized by their rapidly increasing nature. They are the inverse of logarithmic functions and hold foundational properties that allow them to transform logarithmic relationships. For example:

• The equation \(b^{y} = x\) mirrors the operation of applying a logarithm \(\log_b x = y\).
• Exponential functions adhere to laws such as \(b^{m+n} = b^m \times b^n\) and \(b^{m-n} = \frac{b^{m}}{b^{n}}\).

Highlighting exponential function properties is crucial for grasping why logarithms work the way they do, particularly in equations where exponents are involved. Earlier in the exercise, understanding these properties helps to see the connection that \(b^{\log_b x} = x\) simplifies directly back to \(x\) thanks to the interaction between exponential functions and logarithms working as inverses.