Problem 91 - College Algebra

Question

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \log _{4}\left(2 x^{3}\right)=3 \log _{4}(2 x) $$

Step-by-Step Solution

Answer

The given equation \( \log _{4}\left(2 x^{3}\right)=3 \log _{4}(2 x) \) is true.

Step 1: Analyze the given equation

The given equation is \( \log _{4}\left(2 x^{3}\right)=3 \log _{4}(2 x) \). It shows a logarithm of a product on the left side and a multiple of a logarithm on the right side.

Step 2: Apply the logarithmic property to the left side of the equation

Apply the logarithm property \(\log_b (M \cdot N) = \log_b M + \log_b N\) and \(\log_b (M^p) = p \log_b M\) into the left side of equation. The left side of equation becomes \(\log _{4}(2) + \log _{4}(x^{3}) = \log _{4}(2) + 3\log _{4}(x)\)

Step 3: Compare the transformed equation with the right side of the original equation

After transformation, the resulting equation is \(\log _{4}(2) + 3\log _{4}(x) = 3\log _{4}(2 x)\). The right side of the transformed equation is the same as the right side of the original equation. Hence, the given equation is true.

Step 4: Verify the result

In order to make sure whether the conclusion is right or wrong, let's take a specific value of x and check whether both sides of equation are equal or not. Let's take \(x=1\). When we substitute \(x=1\) in both sides of equation, it becomes \[\log_4 2 + 3 \log_4 1 = 3 \log_4 2\]. Simplifying, we find that both sides of the equation are the same and hence the given equation is true.

Key Concepts

Logarithm Properties

Logarithms are powerful tools in algebra, often used to simplify complex calculations, particularly for exponential expressions. Understanding their properties allows us to manipulate and solve equations efficiently.
One crucial property of logarithms is their ability to transform products into sums, expressed as \( \log_b (M \cdot N) = \log_b M + \log_b N \). This is particularly useful in breaking down products within logs into more manageable terms.
Another essential property is the power rule, \( \log_b (M^p) = p \log_b M \), which lets you pull down exponents as coefficients.

• This makes it easier to manage and equate expressions mathematically.
• It can help convert complicated logarithmic expressions into sums and products, easing the process of solving equations.

In the exercise, these properties were used to simplify and verify that both sides of the given logarithmic equation are equivalent. Understanding these principles is key to solving and verifying logarithmic equations correctly.

Exponential Functions

Exponential functions form the basis for logarithmic functions. They are characterized by their growth or decay rates, which occur exponentially.
In mathematical terms, an exponential function is expressed as \( f(x) = a^x \), where \(a \) is a positive constant.
Logarithms are essentially the inverse of these functions, allowing us to find the exponent that a given number must be raised to in order to yield another number.

• This is particularly helpful because it lets us reverse exponential growths or decays, turning them into linear statements that are easier to understand and analyze.
• It is through the understanding of these inverse relationships that many scientific and statistical models operate.

Within the exercise, the use of logarithmic properties engaged with these fundamental ideas of exponential growth to verify the given equation successfully.

Equation Verification

Verifying equations is a fundamental process in mathematics to ensure accuracy and correctness.
To verify an equation, we substitute specific values and check whether both sides of the equation are identical. This process confirms whether the equation holds true under these conditions.
In the given exercise, equation verification was achieved by substituting \( x = 1 \) into the equation, providing a practical way to see the truth of the equation.

• Verification is important as it provides confidence in mathematical results and ensures calculations are correctly implemented.
• By trying out different values, one can verify results beyond mere algebraic manipulation, adding an extra layer of validation to conclusions drawn.

By thorough verification, mathematical conclusions become more robust and trustworthy, aiding in problem-solving and mathematical analysis.