Problem 56 - Algebra A Combined Function

Question

Solve. $$ \log _{9} x=\frac{1}{2} $$

Step-by-Step Solution

Answer

The value of \( x \) is 3.

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_9 x = \frac{1}{2} \). This means that the base 9 logarithm of \( x \) is equal to \( \frac{1}{2} \). We need to find the value of \( x \) such that this equation holds true.

Step 2: Convert Logarithmic to Exponential Form

To solve the logarithmic equation, convert it to its exponential form. The equation \( \log_b a = c \) can be rewritten as \( b^c = a \). Applying this to our equation, we have \( 9^{\frac{1}{2}} = x \).

Step 3: Simplify the Exponential Expression

The expression \( 9^{\frac{1}{2}} \) represents the square root of 9. Simplifying it gives \( \sqrt{9} = 3 \).

Step 4: Conclusion

The value of \( x \) that satisfies the original equation is \( x = 3 \). Thus, the solution is complete.

Key Concepts

Exponential Form

When solving logarithmic equations, it's often helpful to convert the equation into exponential form. This is a fundamental step in understanding and solving log equations. In general, the logarithmic expression \( \log_b a = c \) can be rewritten into exponential form as \( b^c = a \). Here, \( b \) is the base, \( a \) is the result, and \( c \) is the exponent.
By converting to this form, you transform the equation from a more abstract logarithmic expression to a number that can be more easily evaluated.
In our exercise, we begin with \( \log_9 x = \frac{1}{2} \). The exponential form transition is straightforward: think about what power you have to raise 9 to get \( x \). You will rewrite it as \( 9^{\frac{1}{2}} = x \).
This conversion is the backbone of solving the equation, making the unknown more visible and manageable.

Simplifying Expressions

Once you have converted the logarithmic equation into exponential form, the next step is simplification. Simplifying expressions involves breaking down more complex expressions into their most straightforward form. This often involves arithmetic operations, such as multiplication, division, or even extracting roots.
In our case, we simplified the exponential expression \( 9^{\frac{1}{2}} \). The fraction \( \frac{1}{2} \) in the exponent indicates the square root. So, \( 9^{\frac{1}{2}} \) is the same as \( \sqrt{9} \). Simplifying further, \( \sqrt{9} = 3 \).
Simplifying is crucial because it reduces the complexity of math problems, turning difficult operations into basic calculations that are easier to understand and solve. This is particularly important in logarithmic and exponential contexts.

Square Root

Understanding the concept of a square root is vital in solving equations like the one in our exercise. A square root is a number which, when multiplied by itself, gives the original number.
In this exercise, we need to find \( \sqrt{9} \). The square root symbol \( \sqrt{} \) signifies this operation. For example, \( \sqrt{9} \) equals 3 because \( 3 \times 3 = 9 \).
Calculating the square root is a common method to simplify exponential expressions, especially when dealing with fractions as exponents like \( \frac{1}{2} \). Recognizing that an exponent of \( \frac{1}{2} \) implies a square root is an essential aspect of solving logarithmic and exponential problems effectively.

• This understanding helps us tackle various problems with confidence.
• Without this knowledge, solving equations such as this could be confusing.
• So remember: square roots are just as important in math as multiplication and addition.