91Ó°ÊÓ

The properties of logarithms are useful mathematical tools that can help simplify and solve equations. In this exercise, we noticed the application of one key property called the **Quotient Rule**. This rule states:

• For any real numbers \(B\) and \(C\), with the same base \(a > 0\), \(\log_a(B) - \log_a(C) = \log_a\left( \frac{B}{C} \right)\).

By using this rule, we were able to combine the two separate logarithmic terms into a single term:\[ \log_2(8d) - \log_2(2d-1) = \log_2\left( \frac{8d}{2d-1} \right) \]This simplification makes it much easier to work with and helps lay the groundwork for solving the equation. Keeping these properties in mind can help simplify many different kinds of logarithmic expressions and equations.

Once we have simplified the logarithm using the properties of logarithms, the next goal is to solve the equation by eliminating the logarithm. In this particular problem, after using the Quotient Rule, we had:\[ \log_2\left( \frac{8d}{2d-1} \right) = 4 \]To remove the logarithm, you use the property that allows you to rewrite the equation in an exponential form. This property states that if \( \log_a(x) = b \), then \( a^b = x \). Applying it here, we rewrite the equation:\[ 2^{\log_2\left( \frac{8d}{2d-1} \right)} = 2^4 \]The left side simplifies, because \( 2^{\log_2(x)} = x \), which gives:\[ \frac{8d}{2d-1} = 16 \]This exponential step transforms our logarithmic equation into a manageable algebraic one, setting us up perfectly to find the solution for \(d\). Remember, the transformation into an exponential equation is a powerful technique in solving logarithmic equations.

At this stage, algebraic manipulation takes over to find the value of \(d\). With the equation transformed into:\[ \frac{8d}{2d-1} = 16 \]We need to eliminate the fraction by multiplying both sides by \((2d - 1)\):\[ 8d = 16(2d - 1) \]Expand the right side:\[ 8d = 32d - 16 \]The goal here is to isolate \(d\). Begin by moving all terms involving \(d\) to one side and constant numbers to the other:\[ 8d - 32d = -16 \]\[ -24d = -16 \]To solve for \(d\), divide both sides by \(-24\):\[ d = \frac{-16}{-24} = \frac{2}{3} \]And there we have the solution! Understanding how to manipulate algebraic equations is crucial. It involves rearranging and simplifying terms to isolate the variable and solve the equation.