Problem 18 - Precalculus with Limits

Question

Evaluate the function at the indicated value of \(x\) without using a calculator. \(f(x)=\log x \quad x=10\)

Step-by-Step Solution

Answer

The function \(f(x)=\log x\) evaluated at \(x=10\) is \(1\).

Step 1: Understand the properties of logarithms

The logarithm base 10 of a number is the exponent to which 10 must be raised to obtain that number. In other words, \(\log_{10}100 = 2\) because \(10^2 = 100\). Important to note, the base of \(\log\) without a specified base is 10.

Step 2: Apply the value of x into the function

Substitute the given value of \(x\), which is 10, into the function \(f(x)\). So, \(f(10) = \log 10\).

Step 3: Evaluate the function

Using the concept from step 1, we know that \(10^1 = 10\). Thus, \(f(10) = \log 10 = 1\). Thus, the function evaluated at \(x=10\) is \(1\).

Key Concepts

Logarithmic Functions

Logarithmic functions are a type of mathematical function that are the inverse of exponential functions. These functions answer the question: "To what power must a certain number (called the base) be raised, to achieve another number?"
For instance, in the logarithmic equation \( \log_b{y} = x \), \( b^x = y \). This relationship tells us a lot about how logarithms work.

• The function \( \log x \) essentially asks: "What power of 10 is \( x \)?"
• Logarithms are helpful in solving equations where the unknown appears as an exponent.
• They are heavily used in sciences, engineering, and computer science for simplifying calculations.

Understanding the mechanics of logarithmic functions is crucial for evaluating them and applying them to real-world scenarios.

Properties of Logarithms

The properties of logarithms can make computations simpler and aid in solving problems effectively. These properties include rules that are similar to those seen with exponents.

• Product Property: \( \log_b(m \cdot n) = \log_b m + \log_b n \). This means the log of a product is the sum of the logs.
• Quotient Property: \( \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \). This implies the log of a quotient is the difference of the logs.
• Power Property: \( \log_b(m^n) = n \cdot \log_b m \). This means when a number is raised to a power, the power can be brought to the front as a multiplier.
• Change of Base Formula: When changing the base of a logarithm, \( \log_b a = \frac{\log_k a}{\log_k b} \), where \( k \) is a new base.

By understanding these properties, you can solve complex logarithmic equations and greatly simplify otherwise daunting mathematical problems.

Base 10 Logarithms

Base 10 logarithms, often referred to as common logarithms, are logarithms where the base is 10. These are represented simply as \( \log \) without needing to present the base explicitly. Base 10 logarithms are widely used because they are intuitive when dealing with the decimal system we use every day.

• In base 10, \( \log_{10} 10 = 1 \) because \( 10^1 = 10 \).
• If \( \log_{10} 100 = 2 \), this is because \( 10^2 = 100 \), showing how straightforward base 10 can be.
• For values less than 10, such as \( \log_{10} 1 = 0 \), you have \( 10^0 = 1 \), which illustrates how numbers are simplified to powers.

Base 10 logarithms are prevalent in sciences and fields that often involve growth multiplicative processes or phenomena, such as earthquake magnitudes or sound intensity measurements.