Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(10^{\log _{10}(x+5)}\)

Step-by-Step Solution

Answer

The simplified form of the expression \(10^{\log _{10}(x+5)}\) is \(x + 5\)

Step 1 Identifying where to apply the inverse property

On looking at the function \(10^{\log _{10}(x+5)}\), recognize that the base of the exponential function and the base of the logarithmic function are the same (which is 10).

Step 2 Applying the inverse property

Now apply the inverse property of logarithmic function, \(b^{\log_b x} = x\), directly to the function. Replace \(b\) with 10 and \(x\) with (x + 5) to get \(10^{\log _{10}(x+5)} = x+5\).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. This inverse relationship is crucial because it allows you to solve equations involving exponentials by converting them into a more manageable form. When you see a log function like \( \log_b(x) \), it reads "log base \( b \) of \( x \)." What this means is that you are asking the question: "To what power must the base \( b \) be raised to get \( x \)?" Let's break it down further:

• If \( b^y = x \), then \( y = \log_b(x) \).
• The base \( b \) is a constant greater than zero (often 10 or \( e \)), and \( x \) is the value we are dealing with.
• Common logarithms use a base of 10, written as \( \log_{10} \), while natural logarithms use a base of \( e \) and are written as \( \ln \).

Understanding this will help you see how logarithmic functions can simplify complex equations, especially when paired with their exponential counterparts.

Exponential Functions

Exponential functions grow rapidly, defined by equations such as \( b^x \), where \( b \) is a positive constant, often referred to as the base, and \( x \) is the exponent. Exponentials model real-world growth processes like population and compound interest, thanks to their inherent ability to expand quickly over time. Key points to recognize about exponentials include:

• The function \( b^x \) involves raising a base \( b \) to the power of \( x \).
• When \( x \) is positive, the function grows; when \( x \) is negative, the function decreases.
• Exponential functions are often paired with logarithms because of their inverse relationship.

When you encounter exponential functions, recognizing their base and identifying how they can be rewritten using logarithmic functions is crucial, as it simplifies solving equations involving large or small magnitudes.

Simplifying Expressions

Simplifying expressions involves reducing complex mathematical expressions into their simplest form. This can make them easier to work with and understand. In the context of logarithmic and exponential functions, this often involves applying properties such as the inverse property to streamline calculations. For example, using the inverse property of logs and exponentials, \( b^{\log_b(x)} = x \), allows expressions like \( 10^{\log_{10}(x+5)} \) to simplify directly to \( x+5 \).

• Identify the base of the expression and ensure consistency.
• Apply known properties, like the inverse property, to simplify.
• Always seek to bring the equation to its most reduced form, ensuring clarity and ease of further calculations.

Through simplification, complex equations become accessible, paving the way for more efficient problem-solving and deeper comprehension.