Problem 82 - College Algebra

Question

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 3 \log x=\log 125 $$

Step-by-Step Solution

Answer

The exact solution to the equation is \(x = 5\).

Step 1: Simplify the logarithmic equation

Use the properties of logarithms to rewrite the equation. Recall that \(n \log b = \log b^n\), hence we have: \[ 3 \log x = \log 125 \] can be rewritten as: \[ \log x^3 = \log 125 \]

Step 2: Solve for x

When the logarithmic expressions on both sides of an equation are equal, then their arguments must be equal as well. Therefore solve the equation for \(x\): \[ x^3 = 125 \] This solves to \(x = 5\) because \(5^3 = 125\).

Step 3: Check the solution

After obtaining a solution for \(x\), you should always ensure that the obtained solution is in the domain of the original logarithmic expression. Logs are undefined for values less than or equal to 0, but since \(x = 5\) which is greater than 0, this value is valid.

Key Concepts

Properties of Logarithms

Logarithms have unique properties that make solving equations simpler. One of the properties is the "power rule," which states:

• For any positive number \(b\), and any number \(n\), \(n \log b = \log b^n\). This property is used to move coefficients to the exponent, consolidating terms.
• For example, in our original equation \(3 \log x = \log 125\), you can use the power rule to rewrite it as \(\log x^3 = \log 125\).

Understanding these properties allows you to transform and simplify logarithmic expressions effectively. This helps in progressing to the next steps of solving an equation.

Solving Equations

Logarithmic equations can often be tricky, but with the right approach, they're manageable. In the equation \(\log x^3 = \log 125\), the logs on both sides share the same base (typically 10 in common logarithms). When the logarithmic expressions on both sides of an equation are equal, you set the arguments equal to each other:

• In this case, \(x^3 = 125\).
• The next step is solving for \(x\), which involves finding the value that makes this expression true.

Here, solving \(x^3 = 125\) is straightforward. Recognize that \(5^3 = 125\), which directly gives \(x = 5\). Solving logarithmic equations involves these steps of equating and then solving the resulting algebraic expression.

Domain of Logarithmic Functions

The domain of a logarithmic function is crucial for determining valid solutions. A logarithm, \(\log x\), is only defined for values where \(x > 0\). This means any solution where \(x\) is zero or negative must be disregarded.

• In our example, we first find a solution \(x = 5\).
• Next, check this value against the domain of the original equation \(3 \log x = \log 125\).

Since \(x = 5\) is a positive number, it is within the valid domain. Always consider the domain when solving logarithmic equations to ensure the solution is acceptable.

Exact and Approximate Solutions

When solving logarithmic equations, you'll often find both exact and approximate solutions useful, especially when calculations get complex.

• An exact solution is an answer you derive directly from the equation, without rounding. Here, \(x = 5\) is the exact solution since \(5^3 = 125\).
• Approximate solutions are used when you need a decimal or an easier number to work with, which is especially useful when dealing with non-perfect powers or irrational numbers.

In our example, since \(x = 5\) is exact, no approximation is needed. However, if you were to need a decimal approximation, using a calculator would help, and rounding would typically be to two decimal places. This approach is essential when solutions aren't whole numbers or need practical application.