Problem 29 - Algebra and Trigonometry

Question

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{10} x=2} & {\text { (b) } \log _{5} x=2}\end{array} $$

Step-by-Step Solution

Answer

For (a), \( x = 100 \) and for (b), \( x = 25 \).

Step 1: Understand the Problem

We are given two logarithmic equations: \( \log_{10} x = 2 \) and \( \log_{5} x = 2 \). Our task is to find the value of \( x \) in each case using the definition of logarithms.

Step 2: Apply Logarithm Definition for (a)

In the equation \( \log_{10} x = 2 \), the definition of logarithms states that if \( \log_b y = c \), then \( b^c = y \). Therefore, we have \( 10^2 = x \).

Step 3: Calculate for (a)

Calculate \( 10^2 \). This equals 100. So, \( x = 100 \) for part (a).

Step 4: Apply Logarithm Definition for (b)

In the equation \( \log_{5} x = 2 \), again use the logarithm definition: \( 5^2 = x \).

Step 5: Calculate for (b)

Calculate \( 5^2 \). This equals 25. So, \( x = 25 \) for part (b).

Step 6: Verify Solutions

To ensure the solutions are correct, substitute \( x = 100 \) back into \( \log_{10} 100 \), and \( x = 25 \) back into \( \log_{5} 25 \). Both should simplify to 2, confirming our results.

Key Concepts

Logarithms

Logarithms are the inverse operation of exponentiation. If you think of exponentiation as repeated multiplication, logarithms tell us how many times we need to multiply a base to get a certain number. In simpler terms, a logarithm answers the question: "To what power must the base be raised, in order to obtain a specific number?"

• If you have an equation such as \( \log_{b} y = c \), it means that \( b^c = y \).
• For example, \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \).

Understanding this definition is crucial when solving logarithmic problems. It might seem a bit abstract at first, but once you see the connection to exponentiation, it gets easier to see why logarithms are so useful.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base \( b \) and an exponent \( c \). It can be written in the form \( b^c \). When you raise a base to an exponent, you multiply the base by itself repeatedly. For example, \( 10^2 \) means you multiply 10 by itself once, resulting in 100.

• In the logarithmic context, exponentiation helps us "undo" the logarithm operation.
• If you know \( \log_{b} y = c \), you can find \( y \) by calculating \( b^c \).

Consider real-world examples: if you know how many times you need to fold a piece of paper to reach a certain thickness, you are using exponential growth. Exponentiation is critical for understanding phenomena that exhibit growth patterns, like population growth or compound interest.

Problem-solving steps

When tackling logarithmic equations, following systematic problem-solving steps can help ensure accuracy and ease of understanding.
1. **Understand the Problem**: Identify the type of equation you have. Knowing it's a logarithmic equation frames how you'll approach the solution.
2. **Use Definition of Logarithms**: Translate the logarithmic statement into an exponentiation statement using the definition \( \log_{b} y = c \Rightarrow b^c = y \).
3. **Calculate**: Solve the exponentiation to find the numerical value of \( x \). In the given exercise, substitute and compute \( 10^2 \) and \( 5^2 \) to find corresponding \( x \) values.
4. **Verify**: Retrace your steps by substituting back into the original equation to check for consistency. If \( x = 100 \) satisfies \( \log_{10} x = 2 \), your solution is correct. Repeat for each equation to ensure you haven’t made any errors.
These steps not only verify your answers but also deepen your understanding of the connections between logarithms and exponentiation.