Problem 84 - Algebra and Trigonometry

Question

Solve each logarithmic equation. 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. $$ \log (x+7)-\log 3=\log (7 x+1) $$

Step-by-Step Solution

Answer

The result is \(x = 0.2\) or to two decimal places, \(0.20\).

Step 1: Simplify the Equation

Use the properties of logarithms. The difference of two logarithms with the same base can be written as a single logarithm by dividing the argument. This leads to: \(\log\left(\frac{x+7}{3}\right)=\log(7x+1)\)

Step 2: Dealing with the Logarithms

If two logarithms with the same base are equal, then their arguments are also equal. So with this, we can now solve the equation as: \(\frac{x+7}{3}=7x+1\)

Step 3: Solve for x

By rearranging and simplifying, we get: \(x=\frac{4}{20} = 0.2\)

Step 4: Check the Solution

Substitute \(x = 0.2\) into original equation, if the equation holds, then the solution is correct. Otherwise, the solution must be rejected. After checking, \(x = 0.2\) turns out to be a valid solution, since it does not result in a negative value under the logarithm in the original equation.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial when solving logarithmic equations. These properties are mathematical rules that apply to logarithms, similar to how we manipulate exponents.

One fundamental property is that the logarithm of a quotient is the difference of the logarithms; in formal terms, \[ \log_b\left(\frac{x}{y}\right) \] equals \[ \log_b(x) - \log_b(y) \.\] This property allows us to combine or separate logarithmic terms with the same base when they are being added or subtracted.

Another key property is the logarithm of a product, which states that \[ \log_b(xy) \] can be rewritten as \[ \log_b(x) + \log_b(y)\]. There's also the power rule, which asserts that \[ \log_b(x^n) \] is equal to \[ n\cdot\log_b(x) \.\]

These properties are essential tools for manipulating and eventually solving logarithmic equations by isolating the variable of interest.

When an equation presents logarithms on both sides, as seen in the textbook exercise, we can often use these properties to simplify and solve for the variable more efficiently.

Solving Logarithmic Equations

Solving logarithmic equations involves finding the value of the variable that makes the original equation true. As demonstrated in the provided step-by-step solution, it's important to recognize when you can apply the properties of logarithms to simplify the equation.

Isolating the Logarithmic Function

Start by isolating the logarithmic term if necessary, using algebraic steps such as adding, subtracting, or factoring.

Using Logarithm Properties

Then, implement properties of logarithms to combine or break down the logarithmic expressions. This step is pivotal for reducing the complexity of the problem and allows you to set arguments equal when logs have the same base.

Exponentiation

If one is left with a lone logarithm, exponentiation can be used to 'undo' the logarithm and obtain a solvable algebraic expression.

By following these steps carefully and ensuring the mathematical integrity of each operation, one can find the correct value of the variable that solves the equation.

Domain of Logarithmic Functions

The domain of logarithmic functions is a critical concept when solving logarithmic equations. The domain refers to the set of all possible input values (usually denoted as 'x' values) for which the function is defined.

For all real numbers 'b' greater than zero and not equal to one, the logarithm function \[ \log_b(x) \] is only defined for positive 'x' values. This means that the argument of the logarithm, 'x' in this case, must always be positive. When solving logarithmic equations, any solution that would result in a negative or zero argument for the logarithm must be rejected, as it does not fall within the function's domain.

Checking Solutions

After finding potential solutions algebraically, check them against the original equation to ensure they don't violate the domain restrictions. As in the exercise's solution, substituting \(x = 0.2\) confirms it is a valid solution since it keeps all logarithmic expressions within their domain.

Understanding the domain of logarithmic functions is not just a mathematical formality but a necessary step to verify that the solutions obtained are indeed acceptable within the context of the function being explored.