Simplify Logarithmic Expressions With Change Of Base Formula

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In the realm of mathematics, logarithms serve as a powerful tool for simplifying complex expressions and solving equations. Among the various techniques for manipulating logarithms, the change of base formula and the properties of logarithms stand out as essential tools. In this article, we delve into the intricacies of these techniques, demonstrating their application in rewriting a logarithmic expression as a single logarithm in a specified base.

Understanding the Change of Base Formula

At the heart of our exploration lies the change of base formula, a cornerstone of logarithmic transformations. This formula empowers us to convert logarithms from one base to another, thereby facilitating simplification and comparison of logarithmic expressions. The change of base formula is mathematically expressed as follows:

log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}

where a, b, and c are positive real numbers with b ≠ 1 and c ≠ 1. This formula states that the logarithm of a number a to the base b is equal to the logarithm of a to a new base c, divided by the logarithm of b to the same base c. This transformation is crucial when dealing with logarithms that have different bases, as it allows us to express them in a common base for easier manipulation.

The change of base formula is not merely a mathematical curiosity; it is a practical tool that enables us to tackle logarithmic expressions with diverse bases. By converting all logarithms to a common base, we can effectively combine and simplify them using the properties of logarithms. This technique is particularly valuable when dealing with expressions that involve multiple logarithmic terms with varying bases.

Properties of Logarithms

To further enhance our ability to manipulate logarithmic expressions, we turn to the properties of logarithms. These properties provide a set of rules that govern how logarithms interact with various mathematical operations, including addition, subtraction, multiplication, and division. These properties are not just theoretical constructs; they are the workhorses of logarithmic simplification, enabling us to break down complex expressions into manageable components.

  1. Product Rule: The logarithm of the product of two numbers is equal to the sum of their logarithms. Mathematically, this is expressed as:

    log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y

    This rule allows us to transform a logarithm of a product into a sum of logarithms, which can be particularly useful when dealing with expressions involving multiple factors.

  2. Quotient Rule: The logarithm of the quotient of two numbers is equal to the difference of their logarithms. Mathematically, this is expressed as:

    log⁡b(xy)=log⁡bx−log⁡by\log_b (\frac{x}{y}) = \log_b x - \log_b y

    Similar to the product rule, the quotient rule allows us to convert a logarithm of a quotient into a difference of logarithms, which can simplify expressions involving division.

  3. Power Rule: The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. Mathematically, this is expressed as:

    log⁡b(xp)=plog⁡bx\log_b (x^p) = p \log_b x

    The power rule is particularly useful when dealing with expressions involving exponents within logarithms. It allows us to move the exponent outside the logarithm, simplifying the expression.

Rewriting the Expression: A Step-by-Step Approach

Now, let's apply these principles to rewrite the given expression as a single logarithm in the base e. The expression we aim to simplify is:

log⁡e2x5+log⁡e4x28+log⁡e5x25\log _{e^2} x^5+\log _{e^4} x^{28}+\log _{e^5} x^{25}

Our goal is to express this sum of logarithms as a single logarithm with base e. To achieve this, we will employ the change of base formula and the properties of logarithms in a systematic manner.

  1. Applying the Change of Base Formula:

    The first step involves converting each logarithm to the base e using the change of base formula. This will allow us to combine the logarithms more easily. Let's apply the formula to each term:

    log⁡e2x5=log⁡ex5log⁡ee2\log _{e^2} x^5 = \frac{\log_e x^5}{\log_e e^2}

    log⁡e4x28=log⁡ex28log⁡ee4\log _{e^4} x^{28} = \frac{\log_e x^{28}}{\log_e e^4}

    log⁡e5x25=log⁡ex25log⁡ee5\log _{e^5} x^{25} = \frac{\log_e x^{25}}{\log_e e^5}

    Now, our expression becomes:

    log⁡ex5log⁡ee2+log⁡ex28log⁡ee4+log⁡ex25log⁡ee5\frac{\log_e x^5}{\log_e e^2} + \frac{\log_e x^{28}}{\log_e e^4} + \frac{\log_e x^{25}}{\log_e e^5}

  2. Simplifying the Denominators:

    Next, we simplify the denominators using the property log⁡bbp=p\log_b b^p = p. This property states that the logarithm of a base raised to a power is equal to the power itself. Applying this property, we get:

    log⁡ee2=2\log_e e^2 = 2

    log⁡ee4=4\log_e e^4 = 4

    log⁡ee5=5\log_e e^5 = 5

    Substituting these values into our expression, we have:

    log⁡ex52+log⁡ex284+log⁡ex255\frac{\log_e x^5}{2} + \frac{\log_e x^{28}}{4} + \frac{\log_e x^{25}}{5}

  3. Applying the Power Rule:

    Now, we apply the power rule of logarithms, which states that log⁡b(xp)=plog⁡bx\log_b (x^p) = p \log_b x. This allows us to move the exponents outside the logarithms:

    5log⁡ex2+28log⁡ex4+25log⁡ex5\frac{5 \log_e x}{2} + \frac{28 \log_e x}{4} + \frac{25 \log_e x}{5}

  4. Simplifying the Coefficients:

    We can now simplify the coefficients by performing the divisions:

    52log⁡ex+7log⁡ex+5log⁡ex\frac{5}{2} \log_e x + 7 \log_e x + 5 \log_e x

  5. Combining Like Terms:

    Since all the terms now have the same logarithmic part, log⁡ex\log_e x, we can combine them by adding their coefficients:

    (52+7+5)log⁡ex(\frac{5}{2} + 7 + 5) \log_e x

    To add the coefficients, we need a common denominator. The common denominator for 2, 1, and 1 is 2. So, we rewrite the expression as:

    (52+142+102)log⁡ex(\frac{5}{2} + \frac{14}{2} + \frac{10}{2}) \log_e x

    Adding the fractions, we get:

    292log⁡ex\frac{29}{2} \log_e x

  6. Applying the Power Rule in Reverse:

    Finally, we apply the power rule in reverse to express the result as a single logarithm. This means we move the coefficient back into the logarithm as an exponent:

    log⁡ex292\log_e x^{\frac{29}{2}}

Conclusion

Through a systematic application of the change of base formula and the properties of logarithms, we have successfully rewritten the given expression as a single logarithm in the base e. This process highlights the power of these techniques in simplifying complex logarithmic expressions. By mastering these tools, mathematicians, scientists, and engineers can effectively manipulate logarithms to solve a wide range of problems in various fields.

The journey from a sum of logarithms with different bases to a single logarithm with a common base demonstrates the elegance and versatility of logarithmic transformations. The change of base formula serves as a bridge between different logarithmic scales, while the properties of logarithms provide the rules for navigating the logarithmic landscape. Together, these tools empower us to unravel the complexities of logarithmic expressions and unlock their hidden potential.

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Simplifying Logarithmic Expressions Change of Base Formula and Properties