Problem 62 - Intermediate Algebra

Question

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ \log 3 x=\log 9 $$

Step-by-Step Solution

Answer

The exact solution is \( x = 3 \).

Step 1: Understand Logarithmic Equation

The equation \( \log(3x) = \log(9) \) is given. This equation states that the logarithm of some expression (\(3x\)) is equal to the logarithm of a number (9).

Step 2: Apply Property of Logarithms

One of the properties of logarithms is that if \( \log(a) = \log(b) \), then \( a = b \). This means we can set the arguments of the logarithms equal to each other: \( 3x = 9 \).

Step 3: Solve for x

Now, solve the equation \( 3x = 9 \) by dividing both sides by 3 to isolate \( x \). Thus, \( x = \frac{9}{3} = 3 \).

Step 4: Validate Solution

The exact solution to the equation is \( x = 3 \). Since an approximation is not needed in this case (as the solution is an exact integer), there is no further calculation necessary.

Key Concepts

Logarithm Properties

Logarithm properties allow us to simplify and solve many algebraic equations, especially those involving exponential expressions. One essential property of logarithms states that if two logs are equal, then their arguments must also be equal. For example:

• If \( \log(a) = \log(b) \), then \( a = b \).

This is known as the equality property of logarithms. It is crucial when handling equations like the one from the exercise. By applying this property, we can transform an equality of logarithms into an equality involving their arguments. This particular property is very useful in simplifying equations and finding solutions without needing complex calculations.
Logarithms also have other properties, such as:

• 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: \( \log_b(M^n) = n \cdot \log_b(M) \)

Understanding these properties not only aids in solving logarithmic equations but also enhances overall algebraic manipulation skills.

Solving Equations

The process of solving equations involves finding the value(s) of the variable(s) that satisfy the equation. In the context of logarithmic equations, this often means leveraging the properties of logarithms to simplify the equation. As seen in the original exercise, when given:

• \( \log(3x) = \log(9) \)

The property that \( \log(a) = \log(b) \) implies \( a = b \) is used. Thus, we derive the simpler equation \( 3x = 9 \). This drastically reduces the complexity of our problem and transforms it into a basic algebraic equation.
Once you have the simplified equation, follow standard algebra principles:

• Isolate the variable: In the equation \( 3x = 9 \), divide both sides by 3 to isolate \( x \).
• Solve: You get \( x = \frac{9}{3} = 3 \).

The solution process involves recognizing patterns, applying the right logarithmic properties, and simplifying the equation to find the variable's value.

Exact Solutions

In mathematics, an exact solution refers to a solution that is precise and fully accurate, without approximations. This is what we aim to achieve when solving equations, including logarithmic ones.
In the provided exercise, the solution for the equation \( \log(3x) = \log(9) \) yielded an exact value for \( x \) because it solved to the integer \( 3 \).

• An exact solution means the answer is a defined and specific value, not an approximation or estimate.
• Exact solutions can often be preferred when precision is necessary, such as in algebraic proofs or mathematical demonstrations.

While approximations might be necessary in numerical contexts where values are irrational or require decimal representation, finding an exact solution when possible provides clarity and accuracy.
Numerically expressing values to decimal places, such as four decimal places in some contexts, is an approximation used primarily when exactness is impossible or impractical. In the case of \( x = 3 \), no such approximation is needed since it's already a precise integer.