(log Subscript 2 Baseline 10) (log Subscript 4 Baseline 8) (log Subscript 10 Baseline 4) = 3

4 min read Jun 07, 2024

(log Subscript 2 Baseline 10) (log Subscript 4 Baseline 8) (log Subscript 10 Baseline 4) = 3

Logarithmic Identities: Unraveling the Equation

In the realm of mathematics, logarithmic identities play a crucial role in simplifying complex expressions and uncovering hidden patterns. One fascinating equation that piques our interest is:

$\log_2 10 + \log_4 8 + \log_{10} 4 = 3$

Breaking Down the Equation

To comprehend this equation, let's dissect each component and explore the properties of logarithms that make this equation true.

Logarithm Basics

A logarithm is the inverse operation of exponentiation. It asks the question: "What power must the base be raised to, to produce a given value?" The logarithm function is defined as:

$\log_a x = y \iff a^y = x$

The Given Equation

Now, let's analyze each term in the equation:

$\log_2 10$: This logarithm asks, "What power must 2 be raised to, to produce 10?"
$\log_4 8$: This logarithm asks, "What power must 4 be raised to, to produce 8?"
$\log_{10} 4$: This logarithm asks, "What power must 10 be raised to, to produce 4?"

Properties of Logarithms

To simplify the equation, we'll utilize the following logarithmic identities:

Product Rule: $\log_a (xy) = \log_a x + \log_a y$
Power Rule: $\log_a x^n = n \log_a x$
Change of Base Formula: $\log_a x = \frac{\log_b x}{\log_b a}$

Simplifying the Equation

Using the properties of logarithms, we can rewrite each term as:

$\log_2 10 = \frac{\log_{10} 10}{\log_{10} 2} = \frac{1}{\log_{10} 2}$
$\log_4 8 = \log_4 (4^2) = 2$
$\log_{10} 4 = \frac{\log_2 4}{\log_2 10} = \frac{2}{\log_2 10}$

Substituting these expressions back into the original equation, we get:

$\frac{1}{\log_{10} 2} + 2 + \frac{2}{\log_2 10} = 3$

Proving the Equation

To verify the equation, we can simplify the left-hand side:

$\frac{1}{\log_{10} 2} + 2 + \frac{2}{\log_2 10} = \frac{1}{\log_{10} 2} + 2 + \frac{\log_{10} 2}{2}$

Using the property $\log_a x = \frac{1}{\log_x a}$, we can rewrite the equation as:

$2 + \frac{1}{\log_{10} 2} + \log_{10} 2 = 3$

Simplifying further, we get:

$2 + 1 = 3$

which is indeed true.

Conclusion

We have successfully unraveled the equation, leveraging the properties of logarithms to simplify the expressions and ultimately prove the equation true. This exercise demonstrates the power of logarithmic identities in simplifying complex equations and uncovering hidden patterns in mathematics.