Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(\log _{10} 10^{2 x+3}\)

Step-by-Step Solution

Answer

The simplified expression is \(2x+3\).

Step 1: Identify the Base and Exponent

In the expression, the base of the logarithm and the base of the exponent are both 10. The exponent is the expression \(2x+3\)

Step 2: Apply the Inverse Property

Using the inverse property, which tells us that \( \log_{a}{a^{x}} = x\), this simplifies the expression to \(2x+3\).

Key Concepts

Logarithms

Logarithms are all about finding the power to which a base number must be raised to get a certain value. In simple terms, if you have a logarithm like \( \log_{b}(y) = x \), it means that \( b^x = y \).
This is powerful for reversing a base raised to an exponent. Logarithms are the inverse of exponents, which will be useful as we apply the inverse property.
When dealing with logarithms, the base is critical. For instance, a \( \log_{10} \) indicates the base 10, which is quite common in scientific calculations.

Exponential Functions

Exponential functions are expressions where a base is raised to a power. They are written in the form \( b^x \), where \( b \) is the base and \( x \) is the exponent.
Exponents show how many times to multiply the base by itself. For example, in \( 10^{2} \), 10 is multiplied by itself, resulting in 100.
Exponential growth can happen rapidly as the exponent gets larger, especially with higher bases. Understanding exponential functions is key to recognizing how they can be simplified using logarithms.

Base and Exponent

The base and exponent are crucial components of expressions that involve powers. The base is the number that is being multiplied, and the exponent tells us how many times to multiply the base.
In the expression \( 10^{2x+3} \), 10 is the base, and \( 2x+3 \) is the exponent.
When the base of the logarithm and the base of the exponential function are the same, the expression can be simplified using the inverse property. Recognizing the base is particularly important when simplifying expressions using logarithms.

Simplification

Simplification makes mathematical expressions easier to understand and use. When we simplify \( \log_{10} 10^{2x+3} \), we use the inverse property of logarithms.
The inverse property tells us that \( \log_{a}(a^x) = x \). This means that the expression simplifies directly to \( 2x+3 \).
This property uses the fact that logarithms undo exponentiation when the base is the same, providing a straightforward simplified result. Simplifying not only makes calculations easier but also helps in understanding the relationship between logarithms and exponents.