Problem 41 - College Algebra

Question

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$2 \log _{3}(x+4)=\log _{3} 9+2$$

Step-by-Step Solution

Answer

After checking, we find that this logarithm equation has no viable solutions.

Step 1: Simplify the right side

As per the properties of logarithms, we know that \(\log _{b} a^n = n \log _{b} a\). Therefore, we can simplify the term \(\log _{3} 9\) as \(2 \log _{3} 3 \) which is simply \(2\). So, the equation becomes \(2 \log _{3}(x+4) = 2 + 2\).

Step 2: Equate both sides

Now that we have simplified the equation, we can equate both sides of the equation. Subtracting \(2\) from both sides, it gives us \(2 \log _{3}(x+4) = 2\).

Step 3: Solve for x

Divide both sides of the equation by \(2\), we get \(\log_{3}(x+4) = 1\). Using the definition of logs, this implies \(x+4 = 3^1\). Thus, \(x = 3 - 4 = -1\). However, we must reject this value because it results in \(\log_{3}(x+4)\) being a negative number, and the logarithm does not exist for negative numbers.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial when solving logarithmic equations. Logarithms help us transform multiplicative relationships into additive ones, making them easier to manage.
One key property is the power rule:

• The power rule states that \( \log_b(a^n) = n \cdot \log_b(a) \). This allows us to bring down the exponent as a coefficient, simplifying expressions involving powers.

Another important property is the logarithm of a product:

• \( \log_b(MN) = \log_b(M) + \log_b(N) \). This property transforms a product within a log into a sum of logs.

Simplifying expressions using these properties is often a key step in solving logarithmic equations. Recognizing and applying these rules can drastically simplify complex logarithmic expressions.

Solving Equations

Solving logarithmic equations often involves isolating the logarithmic expression and then using properties of logarithms to simplify. Once simplified, you can convert the logarithmic equation into its exponential form. This allows for straightforward algebraic manipulation.
In the given problem, we began by employing the logarithmic property to rewrite terms, making it possible to equate the transformed expression. After simplification, the equation became \( 2 \log_3(x+4) = 4 \).
Next steps involved:

• Dividing both sides by 2, resulting in \( \log_3(x+4) = 1 \).
• Then, by recognizing that \( \log_b(a) = c \) implies \( b^c = a \), we could solve for \( x+4 \).

Finally, the exponential form was solved to find \( x \). This methodical approach ensures we handle logarithms correctly and arrive at potential solutions.

Domain Restrictions

When dealing with logarithms, it's important to remember that not all values of \( x \) are valid. The domain of a logarithmic function is restricted to positive real numbers; this implies:

• You cannot take the logarithm of a negative number.
• You cannot take the logarithm of zero.

For example, in the equation \( \log_3(x+4) \), the expression inside the log, \( x+4 \), must be greater than zero. This leads us to the condition \( x+4 > 0 \) or \( x > -4 \).
Checking this restriction before finalizing values of \( x \) keeps your solution within the acceptable range of values, preventing undefined logarithmic operations. Evaluating domain restrictions early prevents unnecessary calculations on invalid solutions.

Undefined Logarithms

Logarithms can become undefined when their rules are not adhered to. Specifically, logarithms of non-positive numbers are not defined in the real number system. In our example, we reached a solution of \( x = -1 \).
Plugging \( x = -1 \) into \( \log_3(x+4) \) results in \( \log_3(3) \), which is valid. However, if \( x+4 \) becomes non-positive, such as a solution leading to \( x \leq -4 \), it would make the logarithm undefined.
As such, verifying potential solutions against logarithmic constraints is essential. Not rejecting solutions that break these fundamental rules will lead to incorrect results. Always ensure expressions within a logarithm are strictly positive to keep computations valid and meaningful.