Equivalent Expressions To Log Base B Of 3: Explained

Understanding logarithms can sometimes feel like navigating a maze, but with the right tools and knowledge, you can easily find your way through! In this article, we'll break down the question: "Which of the following expressions are equivalent to log_b(3)?" We will explore each option step by step, using the fundamental properties of logarithms to determine their equivalence. Let's dive in and unravel the logarithmic mystery!

Understanding the Basics of Logarithms

Before we jump into the options, let’s quickly recap what logarithms are all about. A logarithm is essentially the inverse operation to exponentiation. When we write log_b(x) = y, we're asking the question: "To what power must we raise b to get x?" The answer is y. Here, b is the base of the logarithm, and x is the argument. Understanding this fundamental relationship is crucial for simplifying logarithmic expressions and determining equivalencies. Key logarithmic properties will be our guiding stars in this exploration. These include the product rule, the quotient rule, and the power rule, which allow us to manipulate and simplify logarithmic expressions with confidence and precision.

Key Logarithmic Properties to Remember

To effectively evaluate the given options, we need to keep a few essential logarithmic properties in mind:

1. Product Rule: log_b(mn) = log_b(m) + log_b(n). This rule tells us that the logarithm of a product is the sum of the logarithms of the individual factors. For example, log₂(8 * 4) can be rewritten as log₂(8) + log₂(4).
2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n). This rule states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator. An example of this rule in action would be log₃(27 / 9) which can be expressed as log₃(27) - log₃(9).
3. Power Rule: log_b(m^p) = p * log_b(m). The logarithm of a number raised to a power is the power times the logarithm of the number. For instance, log₅(25²) is equivalent to 2 * log₅(25).
4. Change of Base Rule: While not directly used in these examples, it’s a handy property to know: log_a(b) = log_c(b) / log_c(a). This allows us to change the base of a logarithm, making it easier to evaluate or compare expressions. For example, if you have log₂(7) and you want to find its value using a calculator that only has a base-10 logarithm function, you can rewrite it as log₁₀(7) / log₁₀(2).

With these properties in our toolkit, we are well-equipped to tackle the expressions and determine which ones are equivalent to log_b(3). Let's move on to evaluating the given options and see how these rules help us simplify and compare logarithmic expressions effectively.

Evaluating the Given Expressions

Now, let's analyze each of the provided expressions to see if they simplify to log_b(3). We'll apply the logarithmic properties we just discussed to break down each expression and compare it to our target.

Option 1: log_b(1/10) + log_b(30)

• Step 1: Apply the Product Rule: Recall that log_b(m) + log_b(n) = log_b(mn). We can combine these two logarithms into a single logarithm by multiplying their arguments: log_b(1/10) + log_b(30) = log_b((1/10) * 30)
• Step 2: Simplify the Argument: Now, we simplify the multiplication inside the logarithm: log_b((1/10) * 30) = log_b(3)

Therefore, the first expression simplifies to log_b(3), making it a correct equivalent.

Option 2: (1/2)log_b(9)

• Step 1: Apply the Power Rule: The power rule states that p * log_b(m) = log_b(m^p). We can rewrite the expression by moving the coefficient (1/2) as an exponent of the argument: (1/2)log_b(9) = log_b(9^1/2)
• Step 2: Simplify the Argument: The exponent (1/2) indicates a square root. We simplify the square root of 9: log_b(9^1/2) = log_b(√9) = log_b(3)

Thus, the second expression also simplifies to log_b(3), confirming its equivalence.

Option 3: (1/3)log_b(27)

• Step 1: Apply the Power Rule: Similar to the previous option, we use the power rule to move the coefficient (1/3) as an exponent of the argument: (1/3)log_b(27) = log_b(27^1/3)
• Step 2: Simplify the Argument: The exponent (1/3) represents a cube root. We calculate the cube root of 27: log_b(27^1/3) = log_b(³√27) = log_b(3)

This expression also simplifies to log_b(3), making it a valid equivalent.

Option 4: -log_b(1/3)

• Step 1: Rewrite the Negative Sign as a Power: The negative sign in front of the logarithm can be treated as a -1 coefficient. We use the power rule to move this -1 as an exponent of the argument: -log_b(1/3) = log_b((1/3)^-1)
• Step 2: Simplify the Argument: A negative exponent indicates a reciprocal. We take the reciprocal of 1/3: log_b((1/3)^-1) = log_b(3)

Therefore, this expression also simplifies to log_b(3), confirming its equivalence.

Option 5: log_b(24) - log_b(8)

• Step 1: Apply the Quotient Rule: The quotient rule states that log_b(m) - log_b(n) = log_b(m/n). We combine the two logarithms into a single logarithm by dividing their arguments: log_b(24) - log_b(8) = log_b(24/8)
• Step 2: Simplify the Argument: Now, we simplify the division inside the logarithm: log_b(24/8) = log_b(3)

This expression also simplifies to log_b(3), making it a correct equivalent.

Conclusion: All Expressions Are Equivalent

After meticulously evaluating each option using the properties of logarithms, we've discovered that all the given expressions are indeed equivalent to log_b(3). This exercise highlights the power and versatility of logarithmic properties in simplifying and manipulating logarithmic expressions. By applying the product rule, quotient rule, and power rule, we were able to transform each expression into the target form, log_b(3).

Understanding these properties not only helps in solving specific problems but also provides a deeper insight into the nature of logarithms and their applications in various fields, including mathematics, science, and engineering. So, next time you encounter a logarithmic expression, remember these fundamental rules, and you'll be well-equipped to tackle it! For further learning and practice, consider exploring resources like Khan Academy's logarithm section. Happy logarithmic exploring!