Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression without using a calculator.\(\log _{10} 100\)

Step-by-Step Solution

Answer

The value of \(\log_{10} 100\) is 2.

Step 1: Understanding the logarithm

A logarithm is essentially an exponent. In the expression \(\log _{10} 100\), the \(10\) is the base, and \(100\) is the number we are taking the logarithm of. We are essentially asking the question: '10 raised to the power of what number gives us 100?'.

Step 2: Applying the definition of logarithm

Since \(10^1 = 10\) and \(10^2 = 100\), we see that the number we are looking for is 2. Therefore, \(\log_{10} 100 = 2\).

Step 3: Confirmation of solution

It's always a good idea to confirm our solution. If we reverse our logarithm to an exponential, we should have \(10^2 = 100\), which confirms that our solution for \(\log_{10} 100\) is indeed 2.

Key Concepts

Evaluating Logarithmic Expressions

Logarithmic expressions can initially seem intimidating, but with some practice, they become less mysterious. Logarithms ask the question: 'To what power must we raise the base to obtain a certain value?' For instance, when evaluating the expression \[\begin{equation}\log_{10} 100\end{equation}\], we interpret this as 'What power must 10 be raised to, to equal 100?' A helpful strategy to evaluate such expressions without a calculator is to look for familiar patterns. Most commonly, you'll compare the argument of the logarithm to powers of the base. For \log_{10} 100, since you know \[\begin{equation}10^2 = 100\end{equation}\], it's clear that the logarithm equals 2. This method is applicable to other bases and arguments as well. Breaking down the components and comparing them to known values simplifies what can often seem like a daunting task.

Exponential Relationships

Exponential relationships are deeply linked with logarithms and provide insight into a wide range of natural phenomena, from population growth to radioactive decay. In an exponential relationship, a quantity grows or decays at a rate proportional to its current value. The general form of an exponential expression is \[\begin{equation}y = b^x\end{equation}\], where \( b \) is the base and \( x \) is the exponent or power. A notable property of exponential functions is that their rate of growth or decay is constant in terms of percentage, not in absolute terms.For example, the expression \( 10^x = 100 \) can be seen as an exponential relationship where you're looking for the value of \( x \) that makes the equation true. Relating this back to the logarithm, \( x = \log_{10} 100 \), shows how logarithms and exponentials are inverse operations. Understanding this relationship allows you to transition seamlessly between the two, often simplifying complex problems.

Logarithm Properties

There are several key properties of logarithms that can make evaluating logarithmic expressions simpler. Here's a summary of three fundamental properties:

• Product Property: \(\log_b(M \cdot N) = \log_b(M) + \log_b(N)\), which states that the logarithm of a product is the sum of the logarithms.
• Quotient Property: \(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\), indicating the logarithm of a quotient is the difference of the logarithms.
• Power Property: \(\log_b(M^k) = k \cdot \log_b(M)\), which tells us that the logarithm of a power is the exponent times the logarithm of the base.

These properties can simplify complex logarithmic expressions and solve logarithmic equations more easily. For example, if tasked with the expression \(\log_{10}(10^3)\), using the power property, you could simplify this to \(3 \cdot \log_{10}(10)\). Since \(\log_{10}(10)\) is 1 (because 10 to the power of 1 is 10), the expression simplifies to just 3. Leveraging these properties is essential for mastering logarithms.