Question

In Exercises, write the expression as the logarithm of a single quantity. $$\frac{1}{3}\ln (x+1)-\frac{2}{3}\ln (x-1)$$

Short answer

The simplified version of the given log expression is $\ln ((x+1)/(x-1{)}^2{)}^{1/3}$.

Step-by-step solution

Use of the Power Rule for Logarithms

Using the property $lo{g}_b(x^n)=n\ast lo{g}_b(x)$ where $lo{g}_b(x^n)$ can be rewritten as $n\ast lo{g}_b(x)$, we write $\frac{1}{3}\ln (x+1)$ as $\ln (x+1{)}^{1/3}$ and $\frac{2}{3}\ln (x-1)$ as $\ln (x-1{)}^{2/3}$. Our expression now becomes $\ln (x+1{)}^{1/3}-\ln (x-1{)}^{2/3}$.

Apply the subtraction property of logarithms

Subtraction between two logarithms can be converted into a division operation inside the log using the property $lo{g}_b(a)-lo{g}_b(b)=lo{g}_b(a/b)$. Thus, converting, we get $\ln \frac{(x+1{)}^{1/3}}{(x-1{)}^{2/3}}$.

Further Simplification

Let's simplify the expression inside the log to form a single quantity: $\ln ((x+1)/(x-1{)}^2{)}^{1/3}$.

Key concepts

Power Rule for Logarithms

Understanding the Power Rule for logarithms can greatly simplify complex logarithmic expressions. This rule states that $lo{g}_b(x^n)=n\ast lo{g}_b(x)$, where ${}^{\prime }{b}^{\prime }$ represents the base of the log, ${}^{\prime }{x}^{\prime }$ is the argument, and ${}^{\prime }{n}^{\prime }$ is the exponent on the argument. In essence, the Power Rule allows us to move the exponent outside the logarithm as a multiplier.

This property is particularly useful when you need to solve equations involving logarithms or simplify expressions where the logarithm of an exponentiated variable is present. For example, when you have an expression like $\frac{1}{3}\ln (x+1)$, applying the Power Rule transforms it into $\ln ((x+1{)}^{1/3})$. The inverse is also true; if you have an expression like $5\ast lo{g}_b(x)$, you can convert it inside the log as a power to get $lo{g}_b(x^5)$.

Overall, the Power Rule makes logarithms less intimidating and more manageable, especially when dealing with multiple logarithmic terms that need to be combined into a single logarithmic expression. It's a cornerstone in understanding the behavior of logarithms in algebraic operations, providing a straightforward method to decompose or construct logarithmic terms for easier manipulation.

Properties of Logarithms

Logarithms have certain properties that mirror those of exponents, given their inverse relationship. These properties make working with logarithmic expressions much easier and more intuitive. Some of the essential Properties of Logarithms are:

• Product Rule: $lo{g}_b(xy)=lo{g}_b(x)+lo{g}_b(y)$, which states that the log of a product is the sum of the logs.
• Quotient Rule: $lo{g}_b(\frac{x}{y})=lo{g}_b(x)-lo{g}_b(y)$, indicating that the log of a quotient is the difference between the logs.
• Change of Base Formula: Which allows converting a logarithm to a different base for easier computation or comparison.
• Log of One: $lo{g}_b(1)=0$, because any number raised to zero is one, reflecting the definition of logarithms.
• Log of the Base: $lo{g}_b(b)=1$, since any base raised to the power of one is itself.

In the context of simplifying logarithmic expressions, using the Quotient Rule can turn a difference of logs into a single log over a quotient. For example, from the exercise $\frac{1}{3}\ln (x+1)-\frac{2}{3}\ln (x-1)$, through the application of the Power Rule and then using the Quotient Rule, the expression simplifies to $\ln (\frac{(x+1{)}^{1/3}}{(x-1{)}^{2/3}})$. This transformative process not only streamlines the expression but also readies it for further algebraic manipulation or for solving equations.

Logarithmic Expressions

Navigating through Logarithmic Expressions requires us to be familiar with the rules and properties of logarithms. A logarithmic expression involves the log function and usually includes variables and arithmetic within it. Simplifying these expressions is a key algebraic skill, especially when you're looking to solve logarithmic equations or inequalities.

When simplifying logarithmic expressions, the goal is to express the complexity of several log terms into a single log term that captures the essence of the original expression. This approach simplifies calculations and provides a clear pathway to solving for variables. In math, as in the given exercise, we might start with a more intricate form, such as $\frac{1}{3}\ln (x+1)-\frac{2}{3}\ln (x-1)$, and through the application of logarithmic properties, end up with a more coherent form like $\ln (((x+1)/(x-1{)}^2{)}^{1/3})$.

Simplifying logarithmic expressions often involves multiple steps. It's important to apply each property correctly and to recognize which property to use at each stage of the simplification process. Mastery of simplifying these expressions leads to a deeper understanding of how logs are used in higher mathematics, such as calculus and beyond, and is crucial in fields that require the interpretation of exponential growth, such as biology, economics, and physics.