Problem 31 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ 3^{\log _{3}(x-1)} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x-1 \).

Step 1: Understand the Logarithmic Property

The logarithmic property states that if you have an expression in the form \( a^{ ext{log}_a(b)} \), then it simplifies directly to \( b \). This is due to the inverse nature of the exponential and the logarithm functions. In other words, \( a^{ ext{log}_a(b)} = b \).

Step 2: Apply the Logarithmic Property

In our expression, \( a = 3 \) and the expression is \( 3^{ ext{log}_3(x-1)} \). By applying the property \( a^{ ext{log}_a(b)} = b \), we can simplify \( 3^{ ext{log}_3(x-1)} \) to \( x-1 \).

Key Concepts

Logarithmic Properties

One of the most powerful tools in mathematics is the set of properties associated with logarithms. These properties help us simplify and solve various mathematical expressions. The key property highlighted in this exercise is that of reciprocal actions between exponents and logarithms.

• If you have an expression like \( a^{\log_a(b)} \), it simplifies directly to \( b \).
• This simplification happens because the exponential function and the logarithm function are inverses of each other. They "cancel out," leaving just the argument \( b \).

Understanding this property means knowing that with any base \( a \), if you exponentiate it by a logarithm with the same base, you're basically just reverse-engineering to the number \( b \). This property is fundamental in algebra, as it connects the concept of exponentiation directly to its inverse— the logarithm.

Expression Simplification

Expression simplification is crucial for making complex problems manageable. In mathematical simplification, our goal is to convert complicated expressions into simpler, more digestible forms without changing their underlying value.

• A simple form often makes it easier to analyze and evaluate expressions or solve equations.
• By applying logarithmic properties, as demonstrated in the exercise, we can greatly reduce the complexity of expressions.

In our example, using the property \( a^{\log_a(b)} = b \), we simplify \( 3^{\log_3(x-1)} \) to just \( x-1 \). This single-step simplification makes it much clearer that the initial expression is equivalent to \( x-1 \). Simplifying expressions not only saves computational effort but also aids in better understanding.

Inverse Functions

Inverse functions, such as logarithms and exponentials, are central to many areas of mathematics. They solve the problem of 'undoing' each other’s operations.

• The exponential function 'raises' a number to a power, dramatically increasing it.
• The logarithm, acting as an inverse, finds the power to which the base should be raised to obtain the number.

When we talk about functions and their inverses, it's all about reversing processes. If \( y = f(x) \), then the inverse \( f^{-1}(y) = x \). In logarithmic applications, this understanding helps us transition back and forth between an initial value and its exponentiated result efficiently. By comprehending these inverses, we unlock the ability to translate between different mathematical forms seamlessly, enhancing our problem-solving skills significantly.