Problem 38 - Algebra 2

Question

Evaluate each expression. \(\log _{9} 243\)

Step-by-Step Solution

Answer

The value of \( \log_{9} 243 \) is \( \frac{5}{2} \).

Step 1: Understand the Problem

We want to find the value of \( \log_{9} 243 \). This means we are looking for the power to which 9 must be raised to get 243.

Step 2: Rewrite the Expression in Exponential Form

Let \( x = \log_{9} 243 \). This translates to the exponential form \( 9^x = 243 \). We need to express 243 as a power of 9 to solve for \( x \).

Step 3: Express 243 as a Power of 9

First, express 243 as a power of 3. Since \( 243 = 3^5 \), try to express it using 9, which is \( 3^2 \). Evaluate: \((3^2)^{\frac{5}{2}} = 3^5 = 243\).

Step 4: Simplify the Exponential Equation

We have \( 9^x = 243 \) and want to express both sides with the same base, 3. Rewrite 9 as \( 3^2 \): \( (3^2)^x = 3^5 \). Then simplify: \( 3^{2x} = 3^5 \).

Step 5: Solve for x

Since the bases (3) are the same, equate the exponents: \( 2x = 5 \). Solve for \( x \) by dividing both sides by 2: \( x = \frac{5}{2} \).

Step 6: Verify the Solution

Verify that when \( x = \frac{5}{2} \), \( 9^x = 243 \). Substitute back: \( 9^{\frac{5}{2}} = (3^2)^{\frac{5}{2}} = 3^5 = 243 \). The solution is correct.

Key Concepts

Exponential Form

When working with logarithms, an important concept to understand is converting a logarithmic expression to its exponential form. This involves rewriting a logarithmic equation to better visualize and solve it.
For example, if you have an expression like \( \log_{b}(a) \), you are essentially asking, _"To what power should the base \( b \) be raised to result in \( a \)?"_
To convert this into exponential form, you would set \( x = \log_{b}(a) \). Doing so transforms your equation into \( b^x = a \). From here, solving the equation becomes more intuitive. You can directly work to find \( x \) by expressing \( a \) as a power of \( b \). This conversion is foundational in simplifying and solving logarithmic expressions effectively.

Base Conversion

In solving logarithmic equations, converting the base into a common base is often crucial. This is known as base conversion, and it is useful when you want to rewrite numbers to share the same base.
Take the original expression \( \log_{9} 243 \) as an example. Here, the goal is to express both 9 and 243 using the same base. Let's start with 9, which can be rewritten as \( 3^2 \). Then look at 243. It is equivalent to \( 3^5 \). Thus, each number now shares the base of 3.
Finding a common base simplifies solving, as it lets you compare and equate exponents directly. By expressing both numbers with base 3, you have \( (3^2)^x = 3^5 \). The exponential equation \( 3^{2x} = 3^5 \) emerges, making it easier to solve for \( x \) since the bases are identical.

Evaluate Expressions

Evaluating expressions entails performing operations and simplifications to find a numerical solution. The steps involved require a logical approach.
Take the example of evaluating \( \log_{9} 243 \). Once the expression is written in exponential form \( 9^x = 243 \), and base conversion is applied so both sides share base 3, you end with the equation \( 3^{2x} = 3^5 \).
To evaluate, compare the exponents directly since the bases match. By equating the exponents, you solve \( 2x = 5 \). By dividing both sides by 2, \( x = \frac{5}{2} \).
Finally, verifying your solution by substituting back into the equation ensures accuracy. Ensuring that the exponential equation equals the original value confirms your evaluation has been executed correctly.