Problem 3 - A Brief Introduction to Engineering Computation with MATLAB

Question

Determine the value of each of the following. $$ \log 10^{2}+10^{5} $$

Step-by-Step Solution

Answer

The value of the given expression is approximately \(5.0004\).

Step 1: Start with the given expression

We begin by working with the given expression: \[ \log 10^{2} + 10^{5} \]

Step 2: Compute the values of \(10^2\) and \(10^5\)

Now we compute the values of \(10^2\) and \(10^5\). 10 raised to the power of 2 means multiplying 10 by itself 2 times, and 10 raised to the power of 5 means multiplying 10 by itself 5 times, so we have: \(10^2 = 10 \cdot 10 = 100\) \(10^5 = 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 = 100{,}000\)

Step 3: Replace the exponents with their computed values

Next, we replace the exponents in the expression with their computed values from step 2: \[ \log (100 + 100{,}000) \]

Step 4: Add the numbers inside the logarithm

Now, we can add the numbers inside the logarithm: \[ \log (100{,}100) \]

Step 5: Evaluate the logarithm to base 10

Finally, we evaluate the base 10 logarithm of 100,100. The base-10 logarithm of a number is the power to which 10 must be raised to obtain that number. In other words, we are looking for the exponent for 10 that gives us 100,100. Using a calculator or another tool to find the logarithm, we get: \[ \log_{10}(100{,}100) \approx 5.0004 \] So the value of the given expression is approximately 5.0004.

Key Concepts

Exponents

Exponents are a key part of math, helping us express very large or very small numbers compactly. When we talk about exponents, we often mean the "power" to which a number is raised. For instance, in the expression \(10^2\), 10 is the base and 2 is the exponent.

Exponential Formulas:

• \(a^n\): where \(a\) is the base and \(n\) is the exponent.
• Raising a number like 10 by an exponent such as 2 means multiplying 10 by itself: \(10 \times 10 = 100\).
• In \(10^5\), it means \(10 \times 10 \times 10 \times 10 \times 10 = 100,000\).

Understanding this makes it easier to follow calculations involving powers, allowing us to simplify expressions significantly.

Base-10 Logarithm

A logarithm answers the question: "To what power must the base be raised, to get a particular number?" Specifically, the base-10 logarithm (or common logarithm) uses base 10. It's often written as \(\log_{10}\) or simply \(\log\).

Understanding Base-10 Logarithms:

• \(\log_{10}(100) = 2\) because \(10^2 = 100\).
• \(\log_{10}(1,000) = 3\) since \(10^3 = 1,000\).
• This means we're asking, "10 to what power equals a specific number?"

In the original exercise, we calculated \(\log_{10}(100,100)\) and found the power needed is approximately 5.0004. Logarithms, especially on calculators and computers, help deal with large numbers easily.

Mathematical Expressions

Mathematical expressions are a combination of numbers, symbols, and operators (like addition, subtraction, multiplication, and division). They represent values and relationships. In our exercise, we started with a mathematical expression \(\log 10^2 + 10^5\).

Breaking Down Mathematical Expressions:

• Each part can be calculated separately using the order of operations, where exponents are evaluated before addition.
• Here, \(10^2\) produces 100, while \(10^5\) gives 100,000.
• Combining these results, we find \(100 + 100,000 = 100,100\).

Evaluating expressions helps us solve complex problems step-by-step. It's important to be systematic, ensuring each component is accurately computed before combining them for the final result.