Problem 100 - Intermediate Algebra

Question

Explain, without using Property \(11.4\), why \(4^{\log _{4} 9}\) equals \(9 .\)

Step-by-Step Solution

Answer

By the definition of a logarithm, \(4^{\log_4 9} = 9\).

Step 1: Understanding the Expression

We start with the expression given: \(4^{\log_4 9}\). This is an exponential expression where the base is 4, and the exponent is \(\log_4 9\). Our goal is to demonstrate that this equals 9.

Step 2: Applying Logarithmic Concepts

Recall the definition of a logarithm: \(\log_b a = c\) implies that \(b^c = a\). For \(\log_4 9\), this means 4 must be raised to some power (\(\log_4 9\)) to yield 9.

Step 3: Substituting Back into the Expression

We know from the logarithm definition step that if \(\log_4 9 = x\), then \(4^x = 9\). Substitute \(x\) back to get \(4^{\log_4 9} = 9\), confirming our goal.

Key Concepts

Exponential Expressions

Exponential expressions involve numbers with exponents, which are used to express repeated multiplication. An exponential expression is of the form \( b^n \), where \( b \) is the base, and \( n \) is the exponent. This notation means that \( b \) is multiplied by itself \( n-1 \) times. For example, \( 4^2 = 4 \times 4 = 16 \). These expressions are powerful in mathematics because they can easily express very large numbers or very small fractions.

When dealing with exponential expressions, it is vital to understand how manipulating the base and exponent affects the outcome. The expression \( 4^{\log_4 9} \) is an interesting case that involves logarithms as the exponent. Here,

• The base is \( 4 \).
• The exponent is \( \log_4 9 \), which is a logarithmic expression.

This particular exponential form is designed to demonstrate a property of logarithms and exponents where the answer simplifies directly back to the number within the log expression, which in this case is \( 9 \).

Logarithms

Logarithms are the opposite of exponents. If you know how exponents work, logarithms let you work backward to find the exponent itself. The logarithm \( \log_b a = c \) means that \( b^c = a \). For example, if \( \log_2 8 = 3 \), then \( 2^3 = 8 \). This is useful in solving equations where the variable is in the exponent.

Logarithms can seem tricky because they work on a different principle than straightforward multiplication or addition. They are most helpful when determining how many times one number, the base, must be multiplied by itself to achieve another number. In our specific case of \( \log_4 9 \), the log tells us that \( 4 \) needs to be raised to some power equal to the outcome 9. This power is exactly \( \log_4 9 \).

• Logarithms simplify multiplication into addition, a helpful trait in complex calculations.
• They are widely used in scientific fields for handling exponential growth and decay.

Properties of Exponents

The properties of exponents help simplify expressions and solve equations that involve exponents. These properties are crucial when dealing with exponential expressions and are as follows:

• Product of Powers: \( b^m \times b^n = b^{m+n} \).
• Quotient of Powers: \( b^m / b^n = b^{m-n} \) (assuming \( b eq 0 \)).
• Power of a Power: \( (b^m)^n = b^{m \cdot n} \).
• Zero Exponent Rule: Any non-zero number raised to the zero power is 1, \( b^0 = 1 \).
• Negative Exponent Rule: \( b^{-n} = 1/b^n \).

In our example of \( 4^{\log_4 9} \), a particular property is at play. Given \( b^{\log_b a} = a \), this combines the understanding of exponent rules with logarithms. Essentially, because \( \log_4 9 \) is the power to which base \( 4 \) needs to be raised to get \( 9 \), applying the exponent property directly yields \( 9 \). This property ensures the expression simplifies directly back to the term inside the logarithm.