Simplify Logarithmic And Algebraic Expressions A Comprehensive Guide
In this comprehensive guide, we will delve into simplifying logarithmic and algebraic expressions without the aid of a calculator. We'll break down each step, providing clear explanations and utilizing the fundamental laws of logarithms and exponents. This guide is designed to enhance your understanding of mathematical simplification, making complex problems more manageable.
1.2 Simplifying Logarithmic Expressions
To simplify logarithmic expressions effectively, it is crucial to understand the basic properties of logarithms. These properties allow us to manipulate and combine logarithmic terms, which is essential for simplifying complex expressions. Let's consider the expression:
(logₑ 1 - log₄ 16) / (log₄ 2 - log₄ 32)
Here, we will simplify the expression step by step, applying logarithmic identities to reach the simplest form. First, recall the fundamental logarithmic identities that will be used throughout this simplification:
- logₐ 1 = 0 for any base a
- logₐ aⁿ = n
- logₐ (b/c) = logₐ b - logₐ c
- logₐ bⁿ = n logₐ b
Step 1: Simplify Individual Logarithms
We start by simplifying individual logarithmic terms in the expression. The first term we encounter is logₑ 1. According to the first logarithmic identity (logₐ 1 = 0), the logarithm of 1 to any base is always 0. Therefore, logₑ 1 equals 0.
Next, we consider log₄ 16. We need to find the power to which 4 must be raised to obtain 16. Since 4² = 16, log₄ 16 simplifies to 2. This simplification uses the identity logₐ aⁿ = n.
In the denominator, we have log₄ 2 and log₄ 32. To simplify these, we express 2 and 32 as powers of 4. We know that 2 is the square root of 4, which can be written as 4^(1/2). Thus, log₄ 2 equals 1/2.
For log₄ 32, we express 32 as a power of 4. We know that 32 = 2⁵ = (4^(1/2))⁵ = 4^(5/2). Therefore, log₄ 32 equals 5/2.
Step 2: Substitute Simplified Values
Now that we have simplified each logarithmic term, we substitute these values back into the original expression:
(0 - 2) / (1/2 - 5/2)
This simplifies the expression to:
-2 / (1/2 - 5/2)
Step 3: Simplify the Denominator
Next, we simplify the denominator by subtracting the fractions. We have:
1/2 - 5/2 = (1 - 5) / 2 = -4 / 2 = -2
So, the expression becomes:
-2 / -2
Step 4: Final Simplification
Finally, we divide -2 by -2, which gives us 1. Therefore, the simplified expression is:
1
By systematically simplifying each term using logarithmic identities and then combining the results, we efficiently simplified the complex logarithmic expression to its simplest form, 1.
1.3 Simplifying Algebraic Expressions
Simplifying algebraic expressions often involves applying the rules of exponents and radicals. Let's consider the expression:
(8x²y (6xy³)) / ³√(y³⁶x¹⁸)
Our goal is to simplify this expression by applying the laws of exponents and simplifying radicals. The steps involved are as follows:
Step 1: Simplify the Numerator
The numerator of the expression is 8x²y multiplied by 6xy³. To simplify this, we multiply the coefficients and combine like terms by adding their exponents. We start by multiplying the coefficients 8 and 6, which gives us 48. Next, we combine the x terms: x² multiplied by x equals x^(2+1) = x³. Similarly, for the y terms, y multiplied by y³ equals y^(1+3) = y⁴. Therefore, the simplified numerator is 48x³y⁴.
Step 2: Simplify the Denominator
The denominator involves a cube root: ³√(y³⁶x¹⁸). To simplify this, we take the cube root of each term inside the radical. The cube root of y³⁶ is y^(36/3) = y¹². The cube root of x¹⁸ is x^(18/3) = x⁶. Thus, the simplified denominator is y¹²x⁶.
Step 3: Rewrite the Expression
Now that we have simplified the numerator and the denominator, we rewrite the entire expression:
(48x³y⁴) / (y¹²x⁶)
Step 4: Simplify by Cancelling Terms
To further simplify the expression, we divide the terms in the numerator by the terms in the denominator. We divide 48 by 1 (since there is no coefficient in the denominator), which remains 48. For the x terms, we have x³ divided by x⁶, which equals x^(3-6) = x⁻³. For the y terms, we have y⁴ divided by y¹², which equals y^(4-12) = y⁻⁸. Therefore, the expression simplifies to 48x⁻³y⁻⁸.
Step 5: Express with Positive Exponents
To express the final answer with positive exponents, we rewrite the terms with negative exponents in the denominator. Thus, x⁻³ becomes 1/x³, and y⁻⁸ becomes 1/y⁸. The fully simplified expression is:
48 / (x³y⁸)
By systematically simplifying the numerator and denominator and then applying the rules of exponents, we efficiently reduced the algebraic expression to its simplest form. This step-by-step approach ensures clarity and accuracy in the simplification process.
1.4 Calculating Logarithmic Values
To calculate logarithmic values, understanding the properties of logarithms is essential. Let's consider the equation:
x = (log₂ 8 + log₃ 9) / log₅ √5
We need to find the value of x by simplifying the logarithmic expressions. This involves applying the laws of logarithms to evaluate each term and then performing the arithmetic operations.
Step 1: Simplify Individual Logarithms
We begin by simplifying each logarithmic term individually. The first term is log₂ 8. We need to find the power to which 2 must be raised to obtain 8. Since 2³ = 8, log₂ 8 simplifies to 3. This is a direct application of the logarithmic identity logₐ aⁿ = n.
The next term is log₃ 9. We need to find the power to which 3 must be raised to obtain 9. Since 3² = 9, log₃ 9 simplifies to 2. Again, this uses the logarithmic identity logₐ aⁿ = n.
The final term is log₅ √5. We need to find the power to which 5 must be raised to obtain √5. We know that √5 can be written as 5^(1/2). Therefore, log₅ √5 simplifies to 1/2. This simplification uses the property that the logarithm of a number raised to a power is the power times the logarithm of the number.
Step 2: Substitute Simplified Values
Now that we have simplified each logarithmic term, we substitute these values back into the original equation:
x = (3 + 2) / (1/2)
This simplifies the equation to:
x = 5 / (1/2)
Step 3: Perform Division
To divide by a fraction, we multiply by its reciprocal. So, dividing 5 by 1/2 is the same as multiplying 5 by 2:
x = 5 * 2
Step 4: Final Calculation
Finally, we perform the multiplication:
x = 10
Therefore, the value of x is 10. By simplifying each logarithmic term using the laws of logarithms and then performing the arithmetic, we efficiently calculated the value of x. This systematic approach ensures clarity and accuracy in solving logarithmic equations.
In conclusion, simplifying logarithmic and algebraic expressions requires a solid understanding of the fundamental laws and properties. By breaking down complex problems into smaller, manageable steps, we can efficiently simplify expressions without the need for a calculator. This guide has provided a comprehensive approach to handling such problems, enhancing your mathematical skills and problem-solving abilities.
