Problem 116 - College Algebra and Calculus: An Applied Approach

Question

Complete the proof of the logarithmic property \(\log _{a} \frac{u}{v}=\log _{a} u-\log _{a} v\) Let \(\log _{a} u=x\) and \(\log _{a} v=y\). \(a^{x}=\quad\) and \(\quad a^{y}=\quad\) Rewrite in exponential form. \(\frac{u}{v}=\frac{\underline{\phantom{xx}}}{u} \quad \begin{aligned}&\text { Divide and substitute for } \\\&u \text { and } v .\end{aligned}\) \(=x-y \quad\) Rewrite in logarithmic form. \(\log _{a} \frac{u}{v}=\quad-\) Substitute for \(x\) and \(y\).

Step-by-Step Solution

Answer

The statement is proved to be true. The proof of the property \(\log_a (u/v) = \log_a u - \log_a v\) is true based on the rules of logarithms and exponents.

Step 1: Assign Logarithmic Variables

Given that \(\log_a u = x\) and \(\log_a v = y\), use the logarithmic representation to rewrite both equations in the exponential form. This will give us \(a^x = u\) and \(a^y = v\).

Step 2: Use Exponential Forms

Now, we take the quotient \(u/v\) and substitute the exponential forms. This will yield \(\frac{a^x}{a^y}\). According to the laws of exponents, \(a^x / a^y = a^{x-y}\).

Step 3: Convert to Logarithmic Form

We now convert the result \(a^{x-y}\) back into logarithmic form. According to laws of logarithm, \(a^{x-y}\) equals to \(\log_a (a^{x-y})\). This can now be simplified as \(\log_a (u/v)\), as \(a^{x-y}\) is equal to \(u/v\) from previous steps.

Step 4: Replace the Variables

The final step is to substitute \(x\) and \(y\) with their respective definitions. Replacing \(x\) with \(\log_a u\) and \(y\) with \(\log_a v\) gives us the final result \(\log_a (u/v) = \log_a u - \log_a v\).

Key Concepts

Exponential and Logarithmic Forms

Understanding the relationship between exponential and logarithmic forms is fundamental in solving various mathematical problems. An exponential form is an expression that represents repeated multiplication of a number, known as the base, raised to a power, called the exponent. For example, in the expression \( a^n \times a^m = a^{n+m} \text{,} \) the base \(a\) is being multiplied by itself \(n + m\) times.

In contrast, a logarithm is the inverse of an exponential function. It tells us what power we need to raise a base to in order to obtain a certain number. If \(a^x = b\), then \(\log_a b = x\). The base \(a\) in the logarithmic form is the same as the base in the exponential form, and \(x\) is the exponent to which base \(a\) must be raised to result in \(b\).

These forms are interconnected and can be used to transition from exponential to logarithmic equations and vice versa, which simplifies the process of solving equations involving exponents and logarithms.

Laws of Exponents

The laws of exponents are a set of rules that describe how to handle operations involving exponents. They are essential tools for simplifying expressions and solving equations that involve exponential terms.

Here are some of the key laws:

• Product of Powers: To multiply two powers with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: To divide two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• Power of a Product: To raise a product to a power, raise each factor to the power: \( (ab)^n = a^n \times b^n \).

These laws allow us to manipulate expressions involving exponents systematically and play a crucial role in both algebraic simplification and in more complex calculus operations.

Laws of Logarithms

The laws of logarithms are the rules that govern the operations on logarithmic functions and enable us to simplify complex logarithmic expressions. Similar to the laws of exponents, they are based on the fundamental properties of logarithms.

Some of the primary laws of logarithms include:

• Product Rule: The logarithm of a product is equal to the sum of the logarithms of the factors: \(\log_a (bc) = \log_a b + \log_a c\).
• Quotient Rule: The logarithm of a quotient is equal to the difference in logarithms: \(\log_a (\frac{b}{c}) = \log_a b - \log_a c\), the very property that is explored in the exercise.
• Power Rule: The logarithm of a power is equal to the exponent times the logarithm of the base: \(\log_a (b^c) = c \cdot \log_a b\).

By applying these laws, we can rewrite logarithmic expressions in simpler forms, which is particularly useful when solving logarithmic equations or when integrating or differentiating logarithmic functions in calculus.

Logarithmic and Exponential Equations

When working with logarithmic and exponential equations, we often need to transition between the logarithmic and exponential forms to find solutions. An exponential equation typically involves variables in the exponent, while a logarithmic equation involves variables within a logarithm.

To solve exponential equations, we may need to take the logarithm of both sides of the equation, allowing us to ‘bring down’ the exponent and solve for the variable. Conversely, solving logarithmic equations often requires us to exponentiate both sides to eliminate the logarithm and solve for the variable.

For instance, to solve for \( x \) in \( a^x = b \), we can take the logarithm with base \( a \) on both sides to get \( \log_a a^x = \log_a b \) which simplifies to \( x = \log_a b \). Likewise, if we have \( \log_a x = c \), exponentiation gives us \( a^c = x \).

It’s important to remember that when solutions involve logarithms, they must satisfy the domain of the logarithmic function; the argument of the logarithm must be positive. These processes form the crux of solving logarithmic and exponential equations, ensuring we can find the values of unknowns in a wide array of applications.