Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Express all solutions in exact form. Do not use a calculator. $$2 \log _{3} x=\frac{4}{5}$$

Step-by-Step Solution

Answer

The solution is \(x = 3^{\frac{2}{5}}\).

Step 1: Simplify the Equation

Consider the given equation: \(2 \log_3 x = \frac{4}{5}\). Divide both sides by 2 to isolate the logarithm, which gives: \(\log_3 x = \frac{2}{5}\).

Step 2: Rewrite the Logarithmic Equation in Exponential Form

Recall that if \(\log_b a = c\), then the equation can be rewritten in exponential form as \(b^c = a\). In this case, \(b = 3\), \(a = x\), and \(c = \frac{2}{5}\). So the equation becomes \(x = 3^{\frac{2}{5}}\).

Step 3: Express the Final Solution in Exact Form

The solution \(x = 3^{\frac{2}{5}}\) is already in the correct form as an exact solution. No further simplification is needed.

Key Concepts

Exponential Form

Understanding how to rewrite logarithmic equations as exponential forms is a crucial skill in solving many algebra problems. In general, when you have a logarithmic equation like \( \log_b a = c \), it can be rewritten in an exponential form as \( b^c = a \). Here, \( b \) is the base of the logarithm, \( a \) is what the logarithm is taken of, and \( c \) is the result of the logarithm. This transformation of equations from logarithmic to exponential form allows us to find unknown values by making use of exponential properties.

Consider the step from the original solution: \( \log_3 x = \frac{2}{5} \). We can express this in exponential form, meaning we will find what \( x \) is by equating 3 (the base of the logarithm) raised to the power \( \frac{2}{5} \). Therefore, \( x = 3^{\frac{2}{5}} \). This step highlights the powerful relationship between logarithms and exponents and how one can be converted into the other to simplify and solve equations.

Exact Form

When asked to provide an exact form of a solution, such as in this exercise, it means representing the answer without any decimal approximations. Many mathematical problems and solutions require us to express numbers as fractions or powers, rather than decimal approximations, to retain the precision and clarity of the solution.

For instance, when dealing with logarithmic and exponential equations, the exact form often involves leaving exponents in their fractional form. It provides a clear view of the solution's structure and its exact value. In our exercise, the expression \( 3^{\frac{2}{5}} \) is the exact form of the solution. Writing the solution in this manner also reflects a clear understanding of the relationship between the numbers involved without rounding off or simplifying to a decimal, which would potentially introduce inaccuracies.

Isolate the Logarithm

Isolating the logarithm is a key step when solving logarithmic equations. This process involves rearranging the equation to make the logarithmic expression stand alone on one side of the equation. This simplification allows for easy conversion to exponential form.

In the given problem, you initially have \( 2 \log_3 x = \frac{4}{5} \). Here, isolating the logarithm involves dividing both sides by 2, giving \( \log_3 x = \frac{2}{5} \). This isolation allows us to clearly see the logarithmic equation in a simple form, facilitating the conversion to exponential form. Once isolated, it's straightforward to determine that \( 3^{\frac{2}{5}} = x \).

This straightforward approach ensures you have a proper foundation when converting and solving the equation, providing a clear pathway to the exact solution.