Expand Log Base 343 Of 125

Logarithmic expressions can sometimes appear complex, but with the right understanding of their properties, they become manageable. One such expression is expanding log base 343 of 125. While this might seem daunting at first glance, it’s a straightforward process once you apply the fundamental rules of logarithms.

Understanding Logarithm Expansion

Logarithm expansion is the process of breaking down a complex logarithmic expression into simpler terms. This is achieved by using the various properties of logarithms, such as the product rule, quotient rule, and power rule. The goal is to isolate variables or simplify the overall expression.

In our case, we are looking to expand ${\log_{343}(125)}$. This represents the power to which 343 must be raised to get 125. Recognizing that both 343 and 125 are powers of smaller integers is key to simplifying this expression.

The Role of Bases and Arguments

In the expression ${\log_{b}(a)}$, ${b}$ is the base and ${a}$ is the argument. The expansion process often involves finding common bases or expressing the argument as a power of the base, or vice versa.

For ${\log_{343}(125)}$, we have a base of 343 and an argument of 125. Our task is to express these numbers in terms of simpler, common bases. We know that ${343 = 7^3}$ and ${125 = 5^3}$.

Applying Logarithm Properties for Expansion

To expand ${\log_{343}(125)}$, we can use the change of base formula and the power rule of logarithms.

Change of Base Formula

The change of base formula allows us to convert a logarithm from one base to another. The formula is:

${\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}}$

where ${c}$ is any convenient new base (commonly 10 or e).

Let's use the natural logarithm (base ${e}$, denoted as ${\ln}$) for our expansion:

${\log_{343}(125) = \frac{\ln(125)}{\ln(343)}}$

Now, we substitute our known powers:

${\frac{\ln(5^3)}{\ln(7^3)}}$

Power Rule of Logarithms

The power rule states that ${\log_{b}(a^n) = n \log_{b}(a)}$. Applying this to our expression:

${\frac{3 \ln(5)}{3 \ln(7)}}$ — Saved By The Bell In Las Vegas: A Complete Guide

We can cancel out the 3s:

${\frac{\ln(5)}{\ln(7)}}$

Simplifying Further with Change of Base

Using the change of base formula in reverse, we can express this simplified fraction back into a single logarithm:

${\frac{\ln(5)}{\ln(7)} = \log_{7}(5)}$

So, the expanded form of ${\log_{343}(125)}$ is ${\log_{7}(5)}$.

Alternative Expansion Method (Direct Substitution)

We can also approach this expansion by directly substituting the known powers of the base and argument.

We have ${\log_{343}(125)}$. Let this expression equal ${x}$:

${\log_{343}(125) = x}$

By the definition of a logarithm, this is equivalent to:

${343^x = 125}$

Now, substitute the prime factorizations:

${(7^3)^x = 5^3}$

Using the exponent rule ${(a^m)^n = a^{mn}}$: — NFL Games Tonight: Schedule, Scores & How To Watch

${7^{3x} = 5^3}$

To solve for ${x}$, we can take the logarithm of both sides. Let's use the natural logarithm:

${\ln(7^{3x}) = \ln(5^3)}$

Apply the power rule of logarithms:

${3x \ln(7) = 3 \ln(5)}$

Divide both sides by 3:

${x \ln(7) = \ln(5)}$ — NYC Polls Close: Know The Election Deadline

Now, solve for ${x}$:

${x = \frac{\ln(5)}{\ln(7)}}$

Again, using the change of base formula in reverse, we get:

${x = \log_{7}(5)}$

This confirms that ${\log_{343}(125) = \log_{7}(5)}$, which is the expanded and simplified form.

Practical Applications of Logarithm Expansion

While this specific example might seem abstract, the principles of expanding and simplifying logarithmic expressions are crucial in various fields:

• Solving Exponential Equations: Simplifying logarithms helps in isolating variables when solving equations involving exponents.
• Scientific Research: Logarithms are used in fields like chemistry (pH scale), seismology (Richter scale), and acoustics (decibel scale). Expansion can simplify complex calculations.
• Computer Science: Logarithmic functions are fundamental in analyzing algorithms and data structures. Understanding their properties aids in complexity analysis.
• Finance: Compound interest calculations often involve logarithmic functions, where simplification can make projections clearer.

Our experience shows that students often find the change of base formula particularly useful when dealing with logarithms that have bases not easily recognized as powers of common numbers.

Frequently Asked Questions (FAQ)

Q1: What does it mean to 'expand' a logarithm?

Expanding a logarithm means breaking down a complex logarithmic expression into simpler components using logarithm properties like the product, quotient, and power rules. The goal is typically to isolate variables or simplify the expression for easier calculation or manipulation.

Q2: What are the main properties of logarithms used for expansion?

The primary properties are:

• Product Rule: ${\log_b(xy) = \log_b(x) + \log_b(y)}$
• Quotient Rule: ${\log_b(x/y) = \log_b(x) - \log_b(y)}$
• Power Rule: ${\log_b(x^n) = n \log_b(x)}$
• Change of Base Formula: (\log_b(a) = \frac{\log_c(a)}{\log_c(b)}]

Q3: Why is recognizing powers of numbers important for expansion?

Recognizing numbers as powers (like ${343 = 7^3}$ and ${125 = 5^3}$) allows you to apply the power rule and the change of base formula effectively, leading to significant simplification.

Q4: Can ${\log_{343}(125)}$ be expressed in other expanded forms?

Yes, using the change of base formula with a common logarithm (base 10, denoted ${\log}$) or natural logarithm (base ${e}$, denoted ${\ln}$) gives:

• (\log_{343}(125) = \frac{\log(125)}{\log(343)}]
• (\log_{343}(125) = \frac{\ln(125)}{\ln(343)}]

Further application of the power rule leads to {\frac{3\log(5)}{3\log(7)} = \frac{\log(5)}{\log(7)}\] or \(\frac{3\ln(5)}{3\ln(7)} = \frac{\ln(5)}{\ln(7)}\] which are numerically equivalent to \(\log_7(5)}.

Q5: Is ${\log_7(5)}$ considered an 'expanded' form of ${\log_{343}(125)}$?

Yes, ${\log_7(5)}$ is the simplified and often considered the most useful expanded form. It expresses the original relationship between 343 and 125 in terms of smaller, more fundamental bases (7 and 5).

Q6: What if the numbers were not perfect powers?

If the numbers were not perfect powers, expansion might still be possible using logarithm properties, but the result might not simplify as neatly. The change of base formula would still be applicable, yielding a ratio of logarithms that might require a calculator for a numerical approximation.

Conclusion

Expanding ${\log_{343}(125)}$ reveals a fundamental relationship between powers and logarithms. By recognizing that ${343 = 7^3}$ and ${125 = 5^3}$, and applying the change of base and power rules, we simplify the expression to ${\log_{7}(5)}$. This process not only demystifies complex logarithmic expressions but also highlights their utility in various analytical and scientific contexts. Practice with similar problems can build confidence and proficiency in manipulating logarithmic functions.

To further explore logarithmic manipulations, consider investigating other examples of the change of base formula or how logarithm properties are used in solving differential equations.