Simplify Log Base 7 Of 12th Root Of 7

Let's dive into simplifying the expression $\log _7(\sqrt[12]{7})$. This problem involves understanding logarithms and radicals, and how they interact with each other. We'll break it down step by step to make it super clear. First, understanding logarithms is crucial. A logarithm is essentially the inverse operation to exponentiation. In simple terms, $\log_b(a) = x$ means that $b^x = a$. Here, $b$ is the base of the logarithm, $a$ is the argument, and $x$ is the exponent. Radicals, on the other hand, represent roots of numbers. For example, $\sqrt[n]{a}$ means the $n$-th root of $a$. It's the number that, when raised to the power of $n$, gives you $a$. Now, let’s understand the interplay between logarithms and radicals. Radicals can be rewritten as fractional exponents. Specifically, $\sqrt[n]{a} = a^{\frac{1}{n}}$. This conversion is incredibly useful because it allows us to apply the properties of exponents to simplify expressions involving radicals. Furthermore, the properties of logarithms will also be used. One essential property is $\log_b(a^c) = c \log_b(a)$. This property allows us to bring exponents outside the logarithm as coefficients, which can greatly simplify expressions. This property is particularly useful when dealing with radicals since they can be expressed as fractional exponents. When confronted with an expression like $\log _7(\sqrt[12]{7})$, the initial strategy is to rewrite the radical as a fractional exponent. Then, we will apply the power rule of logarithms to bring the exponent outside the logarithm. Finally, since $\log_b(b) = 1$, the expression can be simplified to its final form. This approach not only simplifies the given expression but also reinforces the understanding of the relationship between logarithms, radicals, and exponents. Remember, practice is key. Work through similar problems to solidify your understanding and build confidence in manipulating these mathematical concepts. Each time you solve a problem, you reinforce the rules and properties, making them easier to recall and apply in future situations. Understanding these basic principles allows for more complex mathematical manipulations later on, making your grasp of mathematical concepts more thorough and insightful. With consistent practice, what once seemed daunting becomes second nature.

Step-by-Step Solution

Here’s how we can simplify $\log _7(\sqrt[12]{7})$:

1. Rewrite the radical as a fractional exponent: The expression $\sqrt[12]{7}$ can be rewritten as $7^{\frac{1}{12}}$. So, our expression becomes $\log _7(7^{\frac{1}{12}})$.

2. Apply the power rule of logarithms: The power rule states that $\log_b(a^c) = c \log_b(a)$. Applying this rule, we get $\frac{1}{12} \log _7(7)$.

3. Simplify: Since $\log_b(b) = 1$, we know that $\log _7(7) = 1$. Therefore, our expression simplifies to $\frac{1}{12} * 1 = \frac{1}{12}$.

So, $\log _7(\sqrt[12]{7}) = \frac{1}{12}$. Understanding the simplification process of logarithmic expressions involving radicals is essential in mathematics. The key lies in converting radicals to fractional exponents and applying the properties of logarithms correctly. Initially, rewrite the radical expression as a fractional exponent. For example, the $n$-th root of $a$ is equivalent to $a$ raised to the power of $1/n$. Subsequently, use the power rule of logarithms, which states that $\log_b(x^y) = y \log_b(x)$. This rule allows you to bring the exponent outside the logarithm as a coefficient. Finally, simplify the expression using the fact that $\log_b(b) = 1$. This step-by-step approach not only simplifies the given expression but also builds a strong foundation in understanding and manipulating logarithmic and exponential functions. Consistent practice with different types of problems helps reinforce these concepts, enabling you to tackle more complex mathematical challenges with confidence. Remember, each problem you solve is an opportunity to deepen your understanding and improve your skills. With time and practice, these operations will become second nature.

Understanding Logarithms

Before we proceed, let's solidify our understanding of logarithms. Logarithms are the inverse of exponential functions. In simpler terms, if we have $b^x = a$, then we can say that $\log_b(a) = x$. Here, $b$ is the base of the logarithm, $a$ is the argument, and $x$ is the exponent. Understanding this fundamental relationship is crucial for manipulating and simplifying logarithmic expressions. For example, $\log_2(8) = 3$ because $2^3 = 8$. Similarly, $\log_{10}(100) = 2$ because $10^2 = 100$. Grasping this concept allows you to quickly evaluate simple logarithmic expressions and understand how logarithms work. Logarithmic functions have several important properties that make them useful in various mathematical contexts. One of the most important properties is the power rule, which we used in the solution above. The power rule states that $\log_b(a^c) = c \log_b(a)$. This rule is incredibly helpful because it allows us to bring exponents outside the logarithm, which can greatly simplify complex expressions. Another useful property is the product rule, which states that $\log_b(mn) = \log_b(m) + \log_b(n)$. This rule allows us to separate the logarithm of a product into the sum of two logarithms. Similarly, the quotient rule states that $\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)$, which allows us to separate the logarithm of a quotient into the difference of two logarithms. Understanding and applying these properties is essential for simplifying logarithmic expressions and solving logarithmic equations. The ability to manipulate logarithms using these rules is a valuable skill in mathematics and is applicable in various fields such as engineering, physics, and computer science. Moreover, logarithms are used extensively in scales that deal with very large or very small numbers, such as the Richter scale for earthquakes and the pH scale for acidity. Mastering logarithms will undoubtedly enhance your problem-solving abilities and open up new avenues for exploring mathematical concepts.

Working with Radicals

Radicals, often denoted by the symbol $\sqrt{}$, represent the roots of numbers. Understanding how to manipulate radicals is essential for simplifying expressions involving them. The most common radical is the square root, denoted as $\sqrt{a}$, which represents the number that, when multiplied by itself, equals $a$. For example, $\sqrt{9} = 3$ because $3 * 3 = 9$. Similarly, the cube root, denoted as $\sqrt[3]{a}$, represents the number that, when multiplied by itself three times, equals $a$. For example, $\sqrt[3]{8} = 2$ because $2 * 2 * 2 = 8$. In general, the $n$-th root of $a$, denoted as $\sqrt[n]{a}$, represents the number that, when raised to the power of $n$, equals $a$. Radicals can also be expressed as fractional exponents, which often simplifies calculations. The $n$-th root of $a$ can be written as $a^{\frac{1}{n}}$. This conversion is particularly useful when working with expressions that involve both radicals and exponents. For example, $\sqrt{a} = a^{\frac{1}{2}}$, $\sqrt[3]{a} = a^{\frac{1}{3}}$, and $\sqrt[4]{a} = a^{\frac{1}{4}}$. Converting radicals to fractional exponents allows us to apply the rules of exponents, which can greatly simplify complex expressions. When simplifying expressions involving radicals, it's often helpful to look for perfect squares, cubes, or other powers within the radical. For example, $\sqrt{25} = \sqrt{5^2} = 5$, and $\sqrt[3]{27} = \sqrt[3]{3^3} = 3$. Identifying and extracting these perfect powers can simplify the expression significantly. Additionally, when dealing with multiple radicals, it's important to follow the order of operations and simplify from the inside out. Understanding radicals is crucial not only for simplifying algebraic expressions but also for solving equations and understanding various mathematical concepts. Radicals appear in many areas of mathematics, including geometry, trigonometry, and calculus. Mastering radicals will undoubtedly enhance your problem-solving abilities and deepen your understanding of mathematical principles. With practice, you will become more comfortable manipulating radicals and applying them in a variety of contexts.

Conclusion

In summary, simplifying $\log _7(\sqrt[12]{7})$ involves converting the radical to a fractional exponent and then applying the power rule of logarithms. This process demonstrates the interconnectedness of exponents, radicals, and logarithms, highlighting the importance of understanding these fundamental concepts in mathematics. The final simplified expression is $\frac{1}{12}$. Further your understanding of logarithms by visiting Khan Academy's Logarithm Section.