Problem 31 - College Algebra

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$\log _{9}(x)=\frac{1}{2}$$

Step-by-Step Solution

Answer

The solution is \( x = 3 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{9}(x) = \frac{1}{2} \). Here, 9 is the base of the logarithm, \( x \) is the argument, and \( \frac{1}{2} \) is the logarithm value. We need to convert this into its exponential form.

Step 2: Convert to Exponential Form

The general form of a logarithmic equation is \( \log_{b}(a) = c \) which is equivalent to \( b^c = a \). Applying this to our equation \( \log_{9}(x) = \frac{1}{2} \), we convert it to exponential form to get \( 9^{\frac{1}{2}} = x \).

Step 3: Simplify the Exponential Equation

Simplify \( 9^{\frac{1}{2}} \). This expression represents the square root of 9. So, \( 9^{\frac{1}{2}} = \sqrt{9} = 3 \).

Step 4: State the Solution

Since \( x = 9^{\frac{1}{2}} \) and we simplified \( 9^{\frac{1}{2}} \) to 3, the solution to the equation is \( x = 3 \).

Key Concepts

Exponential Form

Let's delve into what it means to convert a logarithmic equation into exponential form. A logarithm is simply a way to express an exponent in a different manner. In the equation \( \log_{b}(a) = c \), the base \( b \) is raised to the power \( c \) to yield the result \( a \). Therefore, this equation can be rewritten in its exponential form as \( b^{c} = a \). Understanding this conversion is crucial as it allows us to solve logarithmic equations by reshaping them into a more familiar format of exponents.

For example, converting \( \log_{9}(x) = \frac{1}{2} \) into exponential form, we see that the base 9 raised to the power \( \frac{1}{2} \) results in \( x \). This conversion is a critical step for solving the equation with ease.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. After converting the logarithmic equation \( \log_{9}(x) = \frac{1}{2} \) into its exponential form, we need to determine the value of \( x \).

Here's a simple process:

• Convert the logarithmic equation to exponential form.
• The equation becomes \( 9^{\frac{1}{2}} = x \).
• Next, simplify \( 9^{\frac{1}{2}} \).

Now, let's simplify it further. \( 9^{\frac{1}{2}} \) is equal to the square root of 9, which results in the number 3. So, we find that \( x = 3 \) completes the equation, confirming the solution.

Conversion of Logarithms

Converting logarithms to exponential form may seem tricky at first, but it's really about translating between two mathematical languages. Logarithmic and exponential forms are inherently linked: one expresses the relationship using exponents while the other uses logarithms.

The reason for conversion, particularly in solving equations, is the often simpler manipulation of exponents over logarithms. When we face an equation like \( \log_{9}(x) = \frac{1}{2} \), recognizing it in exponential form helps visually align the equation with basic algebraic principles.

• It helps us identify and isolate the unknown variable \( x \).
• It transforms complex logarithmic properties into a more digestible exponential format.

So, mastering the conversion of logarithms to their exponential counterpart is key to effectively solving many mathematical problems, making what initially seems complex much more straightforward.