Problem 23 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression without using a calculator. $$\log 10^{x+y}$$

Step-by-Step Solution

Answer

The evaluated expression without using a calculator for \(\log 10^{x+y}\) is \(x + y\).

Step 1: Substitute and compute

Plug in given values and perform the arithmetic.

Step 2: State the result

The value is The evaluated expression without using a calculator for \(\log 10^{x+y}\) is \(x + y\)..

Key Concepts

Evaluate Logarithms

Understanding how to evaluate logarithms is crucial for solving a variety of mathematical problems. A logarithm is an exponent to which a base is raised to produce a certain number. When evaluating an expression like \(\text{log}_b{a}\), you are basically asking the question, 'To what power must we raise the base \(b\) to obtain the number \(a\)?' For example, \(\text{log}_2{8}\) asks for the power that 2 must be raised to result in 8. The answer is 3, because \(2^3 = 8\).

To evaluate logarithms without a calculator, you can rely on the knowledge of basic powers and a solid understanding of the properties of logarithms. Sometimes logarithms can be simplified by applying these properties, making them easier to evaluate manually. For instance, a property states that the log of a number at its own base, such as \(\text{log}_b{b}\), is always equal to 1, because any number raised to the power of 1 is itself. Applying this property makes the evaluation of logarithms much smoother.

Logarithm Equations

Logarithm equations can sometimes be intimidating, but understanding a few key concepts can make them more approachable. A logarithm equation is one that involves a logarithm, typically with the variable you're solving for located within the logarithm or as part of the exponent in the argument of the logarithm. For example, solving the equation \(\text{log}_2{x} = 3\) involves finding the value of \(x\) that makes the equation true.

The solution is to rewrite the equation in exponential form, which is \(2^3 = x\), thus \(x = 8\). Key properties of logarithms, such as the Product Rule, Quotient Rule, and Power Rule, are often used in solving logarithm equations. These properties allow you to manipulate and combine logarithmic expressions in ways that make them easier to solve. By gradually transforming the equation using these mathematical tools, you can isolate the variable and find the solution.

Properties of Logarithms

The properties of logarithms are essential tools in simplifying and solving logarithmic expressions and equations. The three primary properties include the Product Rule, which states that \(\text{log}_b(m \times n) = \text{log}_b{m} + \text{log}_b{n}\), the Quotient Rule, which states that \(\text{log}_b(\frac{m}{n}) = \text{log}_b{m} - \text{log}_b{n}\), and the Power Rule, indicating that \(\text{log}_b({m}^k) = k \times \text{log}_b{m}\). These properties can be used to break down complex logarithmic expressions into simpler terms, which can be evaluated more easily.

Another essential property, often used in solving and simplifying, is that \(\text{log}_b{b} = 1\). This is because any base raised to the first power is itself. Additionally, \(\text{log}_b{1} = 0\) because any base raised to the power of zero equals one. Being familiar with these properties can make a significant difference in understanding and solving logarithmic equations, as seen in the example exercise where the property \(\text{log}_b(b^x) = x\) is applied to simplify \(\text{log}_{10}{10^{(x+y)}}\) directly to the exponent, which is \(x + y\).