Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$ 2 \log _{3} x=3 \log _{3} 5 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\sqrt{5}\).

Step 1: Understand the log equation

Given the equation \(2 \log _{3} x = 3 \log _{3} 5\), recognize that both sides involve logarithms with the same base (3). This means we can use properties of logarithms to simplify.

Step 2: Apply the Power Rule

Use the power rule of logarithms \(a \log_b c = \log_b c^a\) to express both sides: - \(2 \log _{3} x = \log _{3} x^2\) - \(3 \log _{3} 5 = \log _{3} 5^3\)

Step 3: Equalize logarithmic expressions

Set the expressions inside the logarithms equal to each other since the bases are the same: \(x^2 = 5^3\). This gives us an equation without logarithms.

Step 4: Solve the equation for x

Calculate \(5^3 = 125\). So, \(x^2 = 125\). To find \(x\), take the square root of both sides to get \(x = \sqrt{125}\). Simplify \(\sqrt{125}\) to \(x = 5\sqrt{5}\).

Step 5: Solution Verification

Verify by substituting \(x = 5\sqrt{5}\) back into the original equation:- Calculate \(\log_3 (5\sqrt{5})^2\) and \(\log_3 125\).- Both simplify to \(3 \log_3 5\), confirming the solution.

Key Concepts

Power Rule of Logarithms

In logarithmic mathematics, the power rule is a handy tool. It helps to simplify expressions where logarithms involve exponents. The power rule states that if you have a logarithm like \(a \log_b c\), you can rewrite it as \(\log_b c^a\).
This rule allows us to "move" the exponent in front of a log inside, raising the original number to that power.

• Example: \(2\log_3 x\) becomes \(\log_3 x^2\).

By converting the expression using the power rule, we can often simplify equations, making them easier to solve. When both sides of an equation share the same base, like in our problem with base 3, this tool is especially useful.

Equation Solving

Equation solving is a method to find the values of variables that make the equation true. When dealing with logarithmic equations, the goal is often to isolate the variable and "remove" the logarithm. This turns our log equation into a simpler, algebraic equation.

• In our task: After applying the power rule, the equation changes from \(2 \log_3 x = 3 \log_3 5\) to \(\log_3 x^2 = \log_3 125\).
• Since bases are equal and we know that \(\log_b (a) = \log_b (b)\) implies \(a = b\), we set the values inside equal, \(x^2 = 125\).

Once you have an algebraic equation, simple methods like square roots can find the solution. Here, solving \(x^2 = 125\) gives us \(x = 5\sqrt{5}\).
Verification of solutions by substitution is also key to ensure accuracy.

Properties of Logarithms

The properties of logarithms include several rules that help in simplifying and solving equations. Some crucial properties include:

• Product Rule: \(\log_b (mn) = \log_b m + \log_b n\)
• Quotient Rule: \(\log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n\)
• Power Rule: \(a \log_b c = \log_b c^a\)

These rules enable the reduction of complex logarithmic expressions into simpler forms.
Applying these properties properly can transform equations for more straightforward manipulation and solving. When dealing with two logarithms with the same base, equating their contents is often a golden strategy. This occurs because log functions are one-to-one, meaning that equal outputs imply equal inputs.