Problem 31 - Algebra and Trigonometry

Question

Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{3} 243=x} & {\text { (b) } \log _{3} x=3}\end{array} $$

Step-by-Step Solution

Answer

(a) \( x = 5 \), (b) \( x = 27 \).

Step 1: Understand the Logarithmic Definition

The definition of the logarithm states that \( \log_{b} a = c \) means \( b^c = a \). In simpler terms, the logarithm \( \log_{b} a \) gives us the power to which the base \( b \) must be raised to obtain \( a \).

Step 2: Solve for x in Part (a)

In part (a), you have \( \log_3 243 = x \). This implies \( 3^x = 243 \). To find \( x \), recognize that \( 243 \) is a power of \( 3 \). Rewriting \( 243 \) as a power of \( 3 \) gives \( 3^5 = 243 \). Therefore, \( x = 5 \).

Step 3: Solve for x in Part (b)

In part (b), you have \( \log_3 x = 3 \), implying \( 3^3 = x \). Calculate \( 3^3 \) which equals \( 27 \). Thus, \( x = 27 \).

Step 4: Verify Solutions

Re-evaluate both parts: for part (a), \( 3^5 = 243 \) confirms the solution. For part (b), \( 3^3 = 27 \) confirms the solution as well. Both calculations show the solutions are correct.

Key Concepts

Power of a Number

Understanding the power of a number is crucial when working with logarithmic functions. The power, also known as the exponent, determines how many times you multiply the base number by itself. For example, when you see the expression \(3^5\), it means you multiply 3 by itself 5 times: \(3 \times 3 \times 3 \times 3 \times 3\).

• Base: The number that is being multiplied. In our example, this is 3.
• Exponent: The number that tells how many times to multiply the base. Here, it is 5.
• Power of a Number: The result of multiplying the base by itself as many times as the exponent indicates.

When you use logarithms, you are often trying to find this exponent, which shows the power relationship between the base and the result. For example, determining \( x \) in \(3^x = 243\) requires recognizing that \(243 = 3^5\). Thus, the power of a number is 5 in this case.

Properties of Logarithms

Logarithms have specific properties that make solving logarithmic equations easier. These properties are rooted in how logarithms convert multiplication into addition and division into subtraction, making complex calculations simpler. Here are some essential properties that often come in handy:

• Product Rule: \( \log_b (mn) = \log_b m + \log_b n \). This rule shows that the logarithm of a product is the sum of the logarithms of the factors.
• Quotient Rule: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \). This states that the logarithm of a quotient is the difference of the logarithms.
• Power Rule: \( \log_b (m^n) = n \log_b m \). Here, the power in the argument becomes a multiplier in front of the logarithm, simplifying exponent calculations.
• Change of Base Formula: \( \log_b a = \frac{\log_k a}{\log_k b} \). This allows you to change the base of a logarithm, which is useful for calculations where the base is not standard.

Understanding these properties helps manipulate logarithmic expressions and solve equations efficiently.

Solving Logarithmic Equations

Solving logarithmic equations often involves converting a logarithmic form into an exponential form, which can make it easier to solve for the unknown variable. Here's how you can tackle such equations:
First, remember the basic logarithmic relationship represented by \(\log_b a = c\), which implies \(b^c = a\). To solve a logarithmic equation, you often need to apply this conversion.

• Identify the Base and Exponent: Look at your logarithmic equation, such as \(\log_3 x = 3\), and convert it to its exponential form: \(3^3 = x\).
• Solve the Exponential Equation: Calculate the power expression, which in this case results in \(x = 27\).
• Verify Your Solution: By inserting the solution back into the original equation, check if the left-hand side equals the right-hand side. If so, your solution is validated.

Working with logarithms boils down to recognizing patterns in numbers to either rewrite them in exponential form or utilize properties of logarithms, both of which streamline the solving process. Practice helps in becoming proficient in unraveling such equations confidently.