91Ó°ÊÓ

Understanding how to convert a logarithmic equation into its exponential form is key to solving logarithmic equations. In simple terms, a logarithmic equation \( \log_{b}{a} = c \) tells us the power to which a base \( b \) must be raised to obtain the number \( a \). When we convert this into exponential form, it becomes \( b^c = a \).

In the given problem, \( \log_{4}(3x + 1) = -2 \), we identify:

• \( b = 4 \)
• \( a = 3x + 1 \)
• \( c = -2 \)

So, the exponential form of this equation is \( 4^{-2} = 3x + 1 \). This conversion is crucial because it transforms the problem into an equation type that may be more familiar and easier to solve for many students.

Once we have the equation in exponential form, the next step is solving for the variable, \( x \). In our problem, the equation simplifies to \( \frac{1}{16} = 3x + 1 \). From here, we perform a series of straightforward algebraic manipulations:

• Subtract 1 from both sides: \( \frac{1}{16} - 1 = 3x \)
• Simplify the left side: \( -\frac{15}{16} = 3x \)
• Divide both sides by 3 to isolate \( x \): \( -\frac{15}{48} = x \)
• Simplify the fraction: \( x = -\frac{5}{16} \)

Each step should be done with care, ensuring that operations are applied correctly to both sides of the equation. Simplifying fractions and maintaining accuracy in calculations make this process clearer and less prone to errors.

Verifying your solution is an important step that confirms the accuracy of your work. To verify \( x = -\frac{5}{16} \), we substitute \( x \) back into the original equation:

Firstly, evaluate \( 3(-\frac{5}{16}) + 1 \), which gives us \( 1 - \frac{15}{16} = \frac{1}{16} \). Now plug this into the logarithmic expression:

\( \log_{4}(\frac{1}{16}) \).

Using the known logarithmic identity \( \log_{b}{(a^n)} = n\log_{b}{a} \), we recognize that \( \log_{4}(\frac{1}{16}) = \log_{4}(4^{-2}) \), which simplifies to \(-2 \).

This confirms our calculated solution, \( x = -\frac{5}{16} \), is indeed correct. Verification is a powerful tool that not only ensures accuracy but also reinforces the understanding of the solution process and logical reasoning behind each problem-solving step.