Problem 86 - College Algebra Essentials

Question

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). \(\log _{b} 81\)

Step-by-Step Solution

Answer

Therefore, \(\log _{b} 81\) can be expressed as \(4C\) in terms of \(A\) and \(C\).

Step 1: Factorization

Express 81 in terms of 3 as \(81 = 3^{4}\). This makes it possible to rewrite the problem as \(\log _{b} (3^4)\).

Step 2: Use Logarithm Power Rule

The Power Rule of logarithms allows to bring the exponent as a multiplier in front. Apply it in our case to get \(4 \cdot \log _{b} 3\). The Power Rule states that \(\log_b (m^n) = n \cdot \log_b m\).

Step 3: Expression in terms of C

Now it is known that \(\log _{b} 3 = C\), now replace \(\log _{b} 3\) in our equation with \(C\) to get \(4C\).

Key Concepts

Logarithm Power Rule

The Logarithm Power Rule is a fundamental property that allows us to simplify logarithmic expressions involving exponents. Imagine you encounter an expression written as \( \log_b (m^n) \). Instead of feeling overwhelmed, consider this rule as a helpful friend that tells you to 'bring down' the exponent and use it as a multiplier in front of the logarithm. In essence, it transforms the expression into \( n \cdot \log_b m \), making the equation easier to manage.

For example, if we know that \( \log_b 3 = C \) from our given information, and we need to evaluate \( \log_b(3^4) \), we apply the power rule. This yields \( 4 \cdot \log_b 3 \), which simplifies to \( 4C \) when we substitute \( C \) for \( \log_b 3 \).

Using this rule not only simplifies expressions but also helps to solve complex logarithmic equations. It is one of the essential tools in understanding the behavior of logarithms when they interact with exponential terms.

Exponential Expressions

Exponential expressions represent repeated multiplication of a number by itself. They are written in the form \( b^n \), where \( b \) is the base and \( n \) is the exponent. Exponents show how many times the base is used as a factor. Exponential expressions often appear in complications involving growth and decay problems, such as in compound interest calculations or population growth models.

Understanding how to work with exponential expressions is crucial when dealing with logarithmic expressions as well, since logarithms are, in essence, the inverses of exponentials. Thus, they share an intimate connection. Recall our problem where the number 81 was recognized as an exponential expression with base 3 and an exponent of 4, namely, \( 3^4 \). This recognition is key in simplifying logarithmic expressions by either breaking them down into more manageable pieces or converting them into logarithmic form to facilitate further calculations.

Logarithmic Expressions

Contrary to exponential expressions, logarithmic expressions represent the power to which a base must be raised to obtain a particular number. The general form is \( \log_b x \), where \( b \) is the base, \( x \) is the number, and the logarithm itself represents the exponent. Logarithms are particularly useful in solving equations where the unknown is an exponent and appear frequently in scenarios that involve measuring the intensity of sound, earthquakes, and light.

In our exercise, we deal with logarithmic expressions by converting them into forms that involve known values such as \( A \) and \( C \) which represent the logarithms of the numbers 2 and 3 to the base \( b \) respectively. As we've done with the expression \( \log_b 81 \), which can be expressed in terms of \( C \) by utilizing the power rule and the relationship \( \log_b 3 = C \) to simplify the expression. This is a clear demonstration of how various properties of logarithms are used to transform and solve logarithmic equations.