Problem 33

Question

In Exercises $31-46,$ write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\log 3+\log x+\log \sqrt{y}$$

Step-by-Step Solution

Verified

Answer

$\log (3xy^{1/2})$

1Step 1: Rewrite Using Logarithmic Properties

Rewriting the sum of logarithms as a product, we have $\log (3 * x * \sqrt{y})$.

2Step 2: Write the Square Root as a Fraction

Rewriting $\sqrt{y}$ as $y^{1/2}$, gives $\log (3x * y^{1/2})$.

3Step 3: Final Simplification

Resolving the multiplication within the logarithm, arrives to $\log (3xy^{1/2})$.

Key Concepts

Logarithmic PropertiesProduct of LogarithmsSimplification of Logarithmic Expressions

Logarithmic Properties

Logarithmic properties are fundamental tools that make it easier to work with complex expressions. They allow us to rewrite and simplify logarithms using well-known rules. For this exercise, the sum of logarithms is particularly useful.

Product Rule: This rule states that the sum of two or more logarithms can be expressed as the logarithm of the product of the numbers. So, $\log a + \log b = \log(ab)$.
Power Rule: Allows a logarithm of a power to be expressed as a product; $\log a^b = b \cdot \log a$.
Quotient Rule: This rule helps in expressing the difference of two logarithms as a division; $\log a - \log b = \log(\frac{a}{b})$.

By recognizing and applying these rules, complex logarithmic expressions can be transformed into simpler, more manageable forms without needing a calculator.

Product of Logarithms

The product of logarithms refers to the transformation of a sum of logarithms into a single logarithmic expression by leveraging the product rule. When you see a series of added logarithms, it often indicates that these can be condensed into one expression.For the given exercise, we have three terms: $\log 3, \log x, $ and $\log \sqrt{y}$.By using the product rule, we can combine these into one logarithm: $\log (3 * x * \sqrt{y})$.

Practical Use: Reducing the sum to a single logarithm simplifies calculations and helps when further simplification or solving is required.

Understanding how to quickly recognize and apply this transformation is essential for tackling a variety of logarithmic problems.

Simplification of Logarithmic Expressions

Simplification is the process of rewriting an expression in its simplest form, which often makes it easier to understand and use. In logarithmic terms, simplifying might mean condensing, factoring, or rewriting using exponential rules.In this exercise, after using the product property: you are left with $\log (3 * x * \sqrt{y})$. The next step involves handling the square root:

Root to Fraction: The square root $\sqrt{y}$ can be written as a power: $y^{1/2}$, leading to $\log (3x \cdot y^{1/2})$.
Final Expression: This becomes $\log (3xy^{1/2})$, a neat combination of constants, variables, and powers.

By applying these steps, you can efficiently simplify logarithmic expressions, making them easier to interpret and solve.

Problem 33

Other exercises in this chapter

Problem 33

The median price of a new home in the United States rose from $\$ 123,000$ in 1990 to $\$ 220,000$ in $2004 .$ Find an exponential function \(P(t)=C e^{t

View solution

Problem 33

State whether each function is one-to-one. $$f(x)=-2 x^{3}+4$$

View solution

Problem 33

Evaluate each expression without using a calculator. $$\log _{4} 4^{x^{2}+1}$$

View solution

Problem 33

Sketch the graph of each function. $$f(x)=-4(3)^{x}+1$$

View solution