Problem 10 - Calculus and its applications

Question

Solve for \(x\). $$ \log _{5} 125=x $$

Step-by-Step Solution

Answer

The value of \( x \) is 3.

Step 1: Understand the Problem

The problem asks us to find the value of \( x \) that satisfies the equation \( \log_{5} 125 = x \). This means we want to express 125 as a power of 5, since a logarithm to a base 'b' gives us the exponent that the base must be raised to, to get the given number.

Step 2: Express 125 as a Power of 5

To solve this problem, recognize that 125 is a power of 5. We find that \( 125 = 5^3 \). Therefore, we can rewrite the equation \( \log_{5} 125 = x \) as \( \log_{5} (5^3) = x \).

Step 3: Apply the Logarithm Rule

Logarithms have a power rule: \( \log_{b} (b^c) = c \). Applying this rule to our equation \( \log_{5} (5^3) = x \), we have \( x = 3 \).

Step 4: Verify the Solution

Because \( \log_{5} 125 = x \) has been rewritten as \( \log_{5} (5^3) = x \) and evaluated to \( x = 3 \), it confirms that 5 raised to the power of 3 gives us 125, verifying our solution that \( x = 3 \).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics, and understanding them is key to solving many problems, including those involving logarithms. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, when we write \(5^3\), we mean that the number 5 is multiplied by itself three times: \(5 \times 5 \times 5\). Here’s why exponents are so useful:

• They simplify multiplication by repeatedly multiplying the same number. Instead of writing \(5 \times 5 \times 5\), we just write \(5^3\).
• They also provide a way to express large numbers compactly. For example, \(10^6\) is much simpler to write than 1,000,000.
• With exponents, you can easily perform calculations involving powers and roots, and they are an essential tool when dealing with scientific notation.

Powers

Powers, often used interchangeably with exponents, describe the result of raising a base number to an exponent. The power indicates the quantity you would get after carrying out the exponentiation on the base. For example, in the expression \(5^3\), 125 is the power because \(5 \times 5 \times 5 = 125\). Some important properties of powers include:

• Commutative property: This property does not apply to exponents, unlike addition and multiplication products. The order in which you operate matters.
• Associative property: For any three positive numbers, \((a^m)^n = a^{mn}\).
• Distributive property: This property is used for combining factors, such as \((a \times b)^m = a^m \times b^m\).

Powers make calculating the results of long multiplications straightforward and help convey the idea of repeated multiplication effectively.

Logarithmic Properties

Logarithms are the inverse operations of exponentiation, which makes them incredibly helpful for solving equations involving exponents. The basic idea of a logarithm is to determine the exponent necessary to raise a base to get a particular number. For example, \(\log_{5} 125\) asks "to what power must 5 be raised to yield 125?" The answer is 3 since \(5^3 = 125\). Several logarithmic properties can make solving complex equations much simpler:

• Product Property: \(\log_{b}(mn) = \log_{b}m + \log_{b}n\).
• Quotient Property: \(\log_{b}\left(\frac{m}{n}\right) = \log_{b}m - \log_{b}n\).
• Power Property: \(\log_{b}(m^n) = n \cdot \log_{b}m\).
• Base Change Formula: \(\log_{b}m = \frac{\log_{k}m}{\log_{k}b}\) for changing to base \(k\).

Understanding these properties allows us to manipulate and solve logarithmic equations more effectively, just as we demonstrated in the solved exercise, converting the problem into a power and verifying the solution straightforwardly.