Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$2 \log _{m} a-3 \log _{m} b^{2}$$

Step-by-Step Solution

Answer

\( \log_m \left( \frac{a^2}{b^6} \right) \)

Step 1: Apply the Power Rule for Logarithms

The power rule for logarithms states that \( n \log_b(x) = \log_b(x^n) \). First, apply this rule individually to both terms. - For \( 2 \log_{m} a \), use the power rule: \( 2 \log_m a = \log_m a^2 \).- For \( 3 \log_{m} b^2 \), also apply the power rule: \( 3 \log_m b^2 = \log_m (b^2)^3 = \log_m b^6 \).

Step 2: Express as a Sum or Difference of Logarithms

Now rewrite the expression substituting the results from the power rule:\[ \log_m a^2 - \log_m b^6 \].

Step 3: Use the Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right) \). Apply this rule to the current expression:\[ \log_m a^2 - \log_m b^6 = \log_m\left(\frac{a^2}{b^6}\right) \].

Key Concepts

Power Rule for Logarithms

The power rule for logarithms is a helpful property that makes manipulating logarithmic expressions easier. This rule states that the logarithm of a power can be calculated by multiplying the exponent with the logarithm of the base. In other words, for any positive real number and any base:\[n \log_b(x) = \log_b(x^n)\]Let's break it down step-by-step:

• Take the exponent "n" outside the log and move it as a coefficient in front of the logarithm.
• Use this property to express complex logarithmic terms in a simplified form.

In our exercise, we apply this rule to expressions like \( 2 \log_m a \). We rewrite it as \( \log_m a^2 \). Similarly, for \( 3 \log_m b^2 \), we convert it to \( \log_m b^6 \). This method works particularly well when you're trying to untangle or neatly organize complicated expressions. By recognizing powers within an expression, you're able to rewrite them in a way that makes other logarithm properties easier to apply later on.

Quotient Rule for Logarithms

Another fundamental property to understand in the world of logarithms is the quotient rule. This property can turn a log expression involving a difference into a single log expression involving a division. The quotient rule is stated as:\[\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\]This tells us that the difference between two logarithms with the same base can be rewritten as the logarithm of the quotient of their arguments.Here's how you can utilize it effectively:

• This rule is especially useful when working with expressions where you need to combine logs into a single term.
• It simplifies the expression and makes further calculations much neater.

In the context of the exercise, after applying the power rule, the expression \( \log_m a^2 - \log_m b^6 \) is present. By applying the quotient rule, it transforms into \( \log_m \left( \frac{a^2}{b^6} \right) \), combining everything into a single, tidy logarithmic expression. This technique is invaluable, helping you work through problems more efficiently.

Logarithmic Expressions

Understanding how to handle logarithmic expressions is crucial for progressing in mathematics and solving complex equations. Logarithmic expressions often appear complicated, but with the right properties such as the power rule and the quotient rule, they can be simplified effectively.When working with logarithmic expressions, consider the following:

• Identify parts of the expression where rules like power or quotient might be used.
• Look for multiplication, division, or exponents, as these are often hinting that simplifications can be made through the application of logarithm properties.

In our specific exercise:- We started with a mixed expression, involving exponents and logs with coefficients.- By carefully applying the power rule and the quotient rule step by step, we were able to transform and simplify the original expression into \( \log_m\left(\frac{a^2}{b^6}\right) \).
Through practice and understanding, you'll gain confidence in simplifying even the most daunting logarithmic expressions. Recognizing these transformations significantly aids in skills across mathematics.