Solve For X: $3 \log_2(x) - 13 = -11$

In the realm of mathematics, solving for an unknown variable, especially within logarithmic equations, often requires a systematic approach. Today, we're going to tackle a specific logarithmic equation: $3 \log _2(x)-13=-11$. Our goal is to find the exact value of $x$, ensuring we evaluate any radicals where possible and leave our final answer as a whole number, a fraction, or an irreducible radical. We'll steer clear of exponents and decimals, aiming for a clean, precise solution. This problem is a fantastic way to reinforce your understanding of logarithmic properties and algebraic manipulation. Let's break it down step-by-step, making sure every move is clear and logical, so you can confidently approach similar problems in the future. We'll start by isolating the logarithmic term and then proceed to convert the logarithmic equation into an exponential one, a key strategy when dealing with logarithms. Remember, the base of our logarithm is 2, which will be crucial in our final conversion step.

To begin solving the equation $3 \log _2(x)-13=-11$, our first objective is to isolate the logarithmic term, $3 \log _2(x)$. We can achieve this by adding 13 to both sides of the equation. This operation is a fundamental step in simplifying equations, allowing us to work with fewer terms. Performing this addition, we get: $3 \log _2(x) - 13 + 13 = -11 + 13$. Simplifying both sides, we arrive at $3 \log _2(x) = 2$. Now that the logarithmic term is on one side, we need to isolate the logarithm itself, $\log _2(x)$. To do this, we will divide both sides of the equation by 3. This gives us: $\frac{3 \log _2(x)}{3} = \frac{2}{3}$. The result is $\log _2(x) = \frac{2}{3}$. This simplified form is essential because it directly relates the logarithm of $x$ to a constant value. It's important to note that at this stage, we have successfully isolated the logarithm, which is a critical milestone in solving for $x$. We have applied basic algebraic operations – addition and division – to transform the original equation into a more manageable form, setting the stage for the next crucial step: converting the logarithmic equation into its equivalent exponential form. This maneuver is the key to unlocking the value of $x$ directly.

With the equation in the form $\log _2(x) = \frac{2}{3}$, we now transition to solving for $x$ by converting this logarithmic equation into its exponential counterpart. The fundamental relationship between logarithms and exponents states that if $\log_b(y) = a$, then $b^a = y$. In our case, the base $b$ is 2, the exponent $a$ is $\frac{2}{3}$, and the value $y$ is $x$. Applying this definition, we can rewrite our equation $\log _2(x) = \frac{2}{3}$ as $2^{\frac{2}{3}} = x$. This conversion is the most pivotal step in solving logarithmic equations, as it directly expresses $x$ in terms of a base and an exponent, allowing us to bypass logarithmic functions altogether. Now we have $x = 2^{\frac{2}{3}}$. The problem specifies that we should evaluate radicals wherever possible and leave answers as whole numbers, fractions, or radicals. The expression $2^{\frac{2}{3}}$ can be rewritten using radical notation. Remember that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. Therefore, $2^{\frac{2}{3}}$ is equivalent to the cube root of $2^2$. Calculating $2^2$ gives us 4. So, $x = \sqrt[3]{4}$. The number 4 does not have a perfect cube root, meaning $\sqrt[3]{4}$ cannot be simplified into a whole number or a simple fraction. Therefore, $\sqrt[3]{4}$ is our exact value for $x$. We have successfully found the solution, adhering to all the constraints given in the problem statement: no exponents, no decimals, and an evaluated radical where possible (in this case, it's an irreducible radical). This final form, $\sqrt[3]{4}$, is the most precise representation of our solution.

In conclusion, the exact value of $x$ for the equation $3 \log _2(x)-13=-11$ is $\sqrt[3]{4}$. We arrived at this solution by systematically applying algebraic principles and the definition of logarithms. First, we isolated the logarithmic term by adding 13 to both sides, yielding $3 \log _2(x) = 2$. Next, we divided both sides by 3 to get $\log _2(x) = \frac{2}{3}$. The crucial step was converting this logarithmic equation into its exponential form, which gave us $x = 2^{\frac{2}{3}}$. Finally, by expressing the fractional exponent as a radical, we found $x = \sqrt[3]{4}$. This form respects the requirement to leave the answer as a radical when it cannot be evaluated further and avoids decimals and exponents. This process underscores the power of understanding logarithmic properties and algebraic manipulation in solving complex-looking equations. For further exploration into the fascinating world of logarithms and their applications, you might find the resources at Wolfram MathWorld to be incredibly insightful and comprehensive.