Problem 20 - Algebra A Combined Function

Question

Solve each equation. $$ \log _{4} 10-\log _{4} x=2 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{5}{8} \).

Step 1: Understanding the Logarithmic Property

The equation given is \( \log_{4} 10 - \log_{4} x = 2 \). Recall the property of logarithms: \( \log_{b} A - \log_{b} B = \log_{b} \left(\frac{A}{B}\right) \). We can apply this property.

Step 2: Applying the Logarithmic Property

Using the property, combine the logarithms into a single logarithm: \( \log_{4} \left(\frac{10}{x}\right) = 2 \).

Step 3: Eliminate the Logarithm

To remove the logarithm, use the fact that if \( \log_{b} A = C \), then \( A = b^C \). Here, this results in \( \frac{10}{x} = 4^2 \).

Step 4: Calculate the Exponent

Calculate \( 4^2 \) to get \( 16 \), so \( \frac{10}{x} = 16 \).

Step 5: Solve for \( x \)

Solve the equation \( \frac{10}{x} = 16 \). Rearrange this to get \( x = \frac{10}{16} \).

Step 6: Simplify the Solution

Simplify \( \frac{10}{16} \) to its lowest terms, which is \( \frac{5}{8} \).

Key Concepts

Properties of Logarithms

In the world of mathematics, understanding the properties of logarithms is fundamental. One of the key properties allows us to simplify equations by combining or breaking apart logarithmic expressions. This property is:

• The subtraction of two logs with the same base results in the log of a fraction: \(\log_{b} A - \log_{b} B = \log_{b} \left(\frac{A}{B}\right)\).

This principle is crucial because it transforms what might seem like a complex expression into a simpler form.
By using this property, we can consolidate logarithmic terms into a single term, making it easier to handle them, especially when solving equations. It enhances our ability to "see" the underlying structure of the expression.

Solving Equations

When it comes to solving logarithmic equations, the aim is to isolate the variable of interest. In our exercise, after applying the logarithm property, we had \( \log_{4} \left(\frac{10}{x}\right) = 2 \).
The key step here is to eliminate the logarithm to solve for \( x \). We do this by converting the logarithmic form into its exponential equivalent.

• If \( \log_{b} A = C \), then \( A = b^C \).

This conversion is vital as it shifts the equation into a form that we can readily manipulate without the complexities imposed by the logarithm. It puts us in a position to directly calculate and solve for \( x \).

Simplification of Fractions

Once we have isolated \( x \) in the equation, as in the step \( \frac{10}{x} = 16 \), the next step is to simply express \( x \) in terms of known quantities.
We can achieve this by rearranging the equation to \( x = \frac{10}{16} \).
Simplification of fractions is often necessary to present solutions in their most concise form. In our example, \( \frac{10}{16} \) can be simplified by finding the greatest common divisor (GCD) of both the numerator and the denominator.

• The GCD of 10 and 16 is 2.

This results in \( \frac{5}{8} \) which is the simplest form of the fraction.
Being able to simplify fractions is a handy skill that improves readability and understanding of mathematical results.

Exponentiation

Exponentiation plays a critical role when we translate a logarithmic equation into a practical form that can be solved. In the given problem, the exponential form was reached with \( \log_{4} \left(\frac{10}{x}\right) = 2 \) transforming into \( \frac{10}{x} = 4^2 \).
The operation involves calculating \( 4^2 \), which is simply multiplying 4 by itself, resulting in 16.

• Here, 4 is the base and 2 is the exponent.
• Exponentiation is critical because it anchors the process of eliminating logarithms and moving towards a solvable equation.

Understanding exponentiation is not only crucial for solving single-step equations like this, but it's also vital for more complex mathematical processes. It is the bridge that smoothly connects the world of logarithms with coefficients and values we can directly compute and analyze.