Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(10^{\log _{10}\left(x^{2}+7 x+10\right)}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^{2}+7 x+10\).

Step 1: Identify the inverse property

The Inverse Property for logarithmic and exponential functions states that for any positive number 'a' (where a≠1), if 'a' is raised to a power equal to the logarithm base 'a' of 'x', then the answer is 'x'. Symbolically, this is represented as \(a^{log_a(x)} = x\).

Step 2: Apply the inverse property

In this exercise, the base 'a' is 10, and the logarithm base 10 of \(x^{2}+7 x+10\) is the power to which 10 is raised. Therefore, we can apply the Inverse Property to simplify \(10^{log_{10}(x^2+7x+10)}\) as \(x^2+7x+10\).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical operations that help to find the exponent as to which a base number must be raised to get a particular value. In simpler terms, if you have an equation like this: \( 10^y = x \), the logarithm helps us find out that the exponent \( y \) is what makes this equation true. A logarithmic function with a base of 10 is known as a common logarithm, often denoted as \( \log_{10}(x) \). When dealing with logarithmic functions, it's helpful to remember some properties:

• Product Property: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Property: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• Power Property: \( \log_b(m^n) = n \cdot \log_b(m) \)

These properties make working with logarithmic expressions much easier to manipulate and simplify. Once you master these, you can solve complex equations through a methodical approach.

Exponential Functions

Exponential functions are another cornerstone of mathematics. These functions involve expressions in which numbers are raised to the power of variables, such as \( y = b^x \), where \( b \) is a constant base. The base \( 10 \) is often used in exponential expressions because it relates directly to the common logarithm. One key characteristic of exponential functions is their continuous growth or decay properties, which make them useful in modeling situations like population growth or radioactive decay. The exponential function has a few important properties:

• Multiplication Property: \( b^{m+n} = b^m \cdot b^n \)
• Power of a Power: \( (b^m)^n = b^{m\cdot n} \)
• Inverse Property: This states that \( a^{\log_a(x)} = x \) for any positive real number \( x \).

Using these properties, we can simplify complex exponential expressions by breaking them down into more manageable parts.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra and beyond, allowing us to write expressions in their simplest form. This process involves reducing complexity without changing the value of the expression. To simplify expressions, you often perform operations that reduce terms or factors. Here are some steps you usually follow:

• Combine like terms, which share the same variable raised to the same power.
• Use distribution to remove parentheses by multiplying each term within the parentheses by the factor outside.
• Apply properties of exponents and logarithms, such as those discussed earlier for logarithmic and exponential functions.

By simplifying an expression, you not only make it more manageable for further calculations but also prepare it for solving—like in the given exercise, where simplifying \( 10^{\log_{10}(x^2+7x+10)} \) to \( x^2+7x+10 \) uses the Inverse Property to streamline understanding and computation.