Problem 10 - Algebra 2

Question

Solve each equation. Check your solutions. \(2 \log _{5} x=\log _{5} 9\)

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Use Properties of Logarithms

We start with the equation: \(2 \log_{5} x = \log_{5} 9\). We will first use the property of logarithms that allows us to bring the coefficient in front of the logarithm as a power. So, the equation becomes: \(\log_{5} (x^2) = \log_{5} 9\).

Step 2: Equate the Arguments

Since the logs on both sides of the equation have the same base, we can equate the arguments: \(x^2 = 9\).

Step 3: Solve for x

Solve the equation \(x^2 = 9\) by taking the square root of both sides. Remember that taking the square root can yield two solutions: \(x = 3\) or \(x = -3\).

Step 4: Check the Solutions

Substitute each value back into the original equation to check for validity. For \(x = 3\), substitute into the original equation: \(2 \log_{5} 3 = \log_{5} 9\). Since both sides are equal, \(x = 3\) is a valid solution. For \(x = -3\), \(\log_{5}(-3)\) is not defined for real numbers, so \(x = -3\) is not a valid solution.

Key Concepts

Understanding the Properties of Logarithms

When solving logarithmic equations, it's essential to use the properties of logarithms to simplify expressions. One handy property is the power rule, which states that the logarithm of a number raised to an exponent can be rewritten by multiplying the exponent by the logarithm of the number. In mathematical terms, \(a \log_b c = \log_b (c^a)\).

This property can transform complicated equations into simpler ones, making the solving process more straightforward. Using this property, fractions in front of the log can be moved to become exponents of the argument, allowing equations to be equated or further simplified.

Steps for Solving Logarithmic Equations

Solving logarithmic equations often begins by applying logarithmic properties to simplify, as we just learned. In our example equation \(2 \log_{5} x = \log_{5} 9\), we use the power property to transform it into \(\log_{5} (x^2) = \log_{5} 9\).

This transformation allows us to drop the logarithm notation, leading us to a more workable form: the equation \(x^2 = 9\). Solving for \(x\) from here, involves finding the values that satisfy the equation. This typically means isolating \(x\), often by taking square roots or using algebraic methods.

The Importance of Square Roots in Solutions

When you encounter the equation \(x^2 = 9\), you’ll need to find the square root of both sides to solve for \(x\).

Taking the square root gives two potential solutions since \(x^2\) can be \((3)^2\) or \((-3)^2\), yielding \(x = 3\) or \(x = -3\). It's crucial to remember there are two potential roots in these cases.

However, not all solutions might be valid in the context of the original logarithmic equation, due to restrictions imposed by logarithms.

Checking Validity of Solutions

Once the potential solutions are found, it's vital to verify them against the original equation. Not all algebraic solutions will be valid in logarithmic contexts. For instance, negative values are often not permissible in logarithmic bases.

Substitute your solutions, \(x = 3\) and \(x = -3\), back into the original equation. The solution \(x = 3\) satisfies our original equation, while \(x = -3\) does not because the logarithm of a negative number is undefined in the real number system. Always check each potential solution to ensure the values fit within the constraints of the logarithmic equation.