Problem 49 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log x+3 \log y $$

Step-by-Step Solution

Answer

The given logarithmic expression is condensed to \(\log(xy^3)\).

Step 1: Apply logarithm properties

The given expression is \(\log x+3 \log y\). From the properties of log, we know that \(a \log b = \log(b^a)\) . So we can simplify the expression as \(\log x+ \log(y^3)\).

Step 2: Simplify using logarithm rules

We can further condense the expression into a single logarithm using the rule that \(\log a + \log b = \log(ab)\). Substituting our values, we get \(\log(xy^3)\). This expression is now a single logarithm with a coefficient of 1.

Step 3: Conclusion

So, the logarithmic expression has been simplified to \(\log(xy^3)\). Where values of x and y are known, the expression above can be evaluated.

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical expressions involving logarithms. A logarithm is essentially the inverse function to exponentiation. In simpler terms, a logarithm finds the power to which a number (known as the base) must be raised to obtain another number. For example, in the expression \( \log_b(x) \), we are determining what power the base \( b \) needs to be raised to get \( x \).

When working with logarithmic expressions, we often encounter terms like \( \log(x+y) \) or \( \log(x) + \log(y) \). Understanding these expressions is crucial as they often appear in various mathematical problems, especially in calculus and algebra.

• A logarithmic expression like \( 3 \log(y) \) means that y is raised to the power of 3, resulting in \( \log(y^3) \).
• The expression \( \log(x) + 3 \log(y) \) can be simplified further by applying properties of logarithms.

Properties of Logarithms

The properties of logarithms are essential tools for manipulating and simplifying logarithmic expressions. They allow us to condense or expand expressions, making them simpler to work with. Here are some of the most useful properties:

• The **Product Property**: \( \log_b(m) + \log_b(n) = \log_b(mn) \). This property tells us that the sum of the logarithms is equivalent to the logarithm of the product of the numbers.
• The **Power Property**: \( a \log_b(x) = \log_b(x^a) \). Here, a coefficient can be moved to the exponent position within the logarithm, which is helpful in simplifying expressions.
• The **Quotient Property**: \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \). It states that the difference of logarithms is the logarithm of a quotient.

In the given expression \( \log(x) + 3 \log(y) \), we use the Power Property to rewrite \( 3 \log(y) \) as \( \log(y^3) \). Then, we apply the Product Property to combine it with \( \log(x) \) yielding \( \log(xy^3) \).

Single Logarithm

The concept of writing an expression as a single logarithm is particularly useful for simplification. When expressions are condensed, they often become easier to interpret and solve. This means taking multiple logarithmic terms and reducing them to a single, compact form.

For instance, if you start with \( \log(x) + 3 \log(y) \), the aim is to combine these into one logarithmic expression. By utilizing the properties explained earlier, you can transform this into \( \log(xy^3) \).

• Firstly, use the Power Property to simplify any coefficients.
• Secondly, apply the Product Property to combine logarithms into a single expression.

This conversion to a single logarithm means the coefficient of the logarithm becomes 1, making it simpler. When expressions are written this way, they are more uniform and often easier to evaluate if the values are known.