Problem 84 - Algebra and Trigonometry

Question

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\)Write each expression in terms of \(A\) and \(C\). $$ \log _{b} 6 $$

Step-by-Step Solution

Answer

\(\log_b 6\) in terms of \(A\) and \(C\) is \(A + C\)

Step 1: Expression of given value

Given that \(\log _{b} 2=A\) and \(\log _{b} 3=C \), the goal is to express \(\log _{b} 6\) in terms of \(A\) and \(C\).

Step 2: Use log rules

Remember that \(\log _{b} (mn) = \log_b m + \log_b n\), meaning that logs of products can be re-written as sums of logs. Therefore, since \(6 = 2 * 3\), we can write \(\log _{b} 6 = \log_b 2 + \log_b 3\)

Step 3: Substitute \(A\) and \(C\)

Substitute the given values into the obtained expression, which means replace \(\log _{b} 2\) with \(A\) and \(\log _{b} 3\) with \(C\) therefore, the expression \(\log _{b} 6\) becomes \(A + C\)

Key Concepts

Logarithmic Identities

Logarithmic identities are powerful tools for simplifying expressions involving logarithms. At the heart of these identities is the concept that logs help break down multiplication, division, and exponentiation into simpler arithmetic. These identities are based on fundamental properties of logarithms including:

• Product Rule: The logarithm of a product is the sum of logarithms. If you have two numbers, say, \(m\) and \(n\), then \( \log_{b}(mn) = \log_{b} m + \log_{b} n \).
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms. This is expressed as \( \log_{b}\left(\frac{m}{n}\right) = \log_{b} m - \log_{b} n \).
• Power Rule: The logarithm of a power is the exponent times the logarithm of the base. This rule can be stated as \( \log_{b}(m^n) = n \cdot \log_{b} m \).

Utilizing these rules allows you to manipulate and simplify logarithmic expressions into a more manageable form. In our exercise, we used the Product Rule to express \( \log_{b} 6 \) as \( \log_{b} 2 + \log_{b} 3 \). By substituting the given values of \( \log_{b} 2 = A \) and \( \log_{b} 3 = C \), the expression simplifies to \( A + C \).

Exponential Functions

Exponential functions are fundamental in mathematics, closely linked with logarithms. Understanding how logarithms relate to exponents can deepen your comprehension of many math concepts. An exponential function can be written generally as \( f(x) = b^x \), where \( b \) is the base and \( x \) is the exponent. Logarithms, conversely, are the inverse functions of exponents.Let's delve into the relationship between these concepts:

• An exponential function like \( b^2 = 4 \) implies that in order to express this in logarithmic form, you would write \( \log_{b} 4 = 2 \).
• Logarithms help find the exponent \( x \) when the base and the output of the exponential function \( b^x \) are known.

The connection between logarithms and exponents is essentially about "undoing" each other. If you take the logarithm of an exponential function, you revert to the original exponent. Understanding this interaction allows you to solve for unknown values and reframe problems involving exponential growth, decay, or any process that follows the principle of repeated multiplication.

Base of a Logarithm

The base of a logarithm is a critical aspect that defines its properties and the outcome of the calculations. It is the number that is raised to the power of the logarithm to produce a given number. When writing the logarithm \( \log_{b} x \), \( b \) is the base, while \( x \) is the number we wish to find the log for.Choosing the correct base is vital:

• Common bases include \( 10 \) (known as the common logarithm) and \( e \) (known as the natural logarithm, \( \ln \)).
• The base establishes the "scale" on which the logarithm operates. For example, \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \).
• When no base is shown, it is generally assumed to be \( 10 \).

In the exercise given, \( b \) serves as the base of all logarithmic expressions involved. Understanding the base allows for consistent application of the logarithmic identities and facilitates accurate substitutions of given known values, like \( A = \log_{b} 2 \) and \( C = \log_{b} 3 \). The base effectively "grounds" the logarithm, allowing it to be manipulated using the mentioned identities to solve and simplify problems effectively.