Problem 2 - Algebra 2 and Trigonometry

Question

Melita said that the equation \(4(3)^{x}=72\) could be solved by writing the logarithmic equation \(x \log 12=\log 72 .\) Do you agree with Melita? Explain why or why not.

Step-by-Step Solution

Answer

No, Melita's equation is incorrect; bases do not match.

Step 1: Analyze the Given Equation

We start with the equation \(4(3)^x = 72\). Here, we want to solve for \(x\). First, we divide both sides of the equation by 4 to simplify it. This yields:\[(3)^x = \frac{72}{4}\]\(\Rightarrow (3)^x = 18\).

Step 2: Align Equation with Logarithmic Form

Next, we want to express this exponential equation in logarithmic form. We know that \(b^y = z\) can be rewritten as \(y = \log_b(z)\). Therefore, expressing \((3)^x = 18\) in logarithmic form, we get:\[x = \log_3(18)\]

Step 3: Compare with Melita’s Logarithmic Equation

Now, consider Melita’s proposed equation: \(x \log 12 = \log 72\). For Melita's equation to correctly represent our transformed equation, we expect both to logically equate to \(x\). First, review \(\log_3(18)\) from Step 2.\[x = \frac{\log 18}{\log 3}\]Breaking down Melita's: \[x = \frac{\log 72}{\log 12}\]Melita’s setup equates to an entirely different base and argument, and it is not equivalent to the standard conversion from \((3)^x = 18\) solved above.

Key Concepts

logarithmic form

When dealing with exponential equations, it is often useful to convert them into a logarithmic form. This conversion simplifies solving for unknowns. Logarithmic form is based on the relationship between exponents and logarithms:

• If you have an exponential equation in the form of \(b^y = z\), it can be rewritten as \(y = \log_b(z)\).
• This means you are solving for the exponent, \(y\), by finding the logarithm of \(z\) with base \(b\).

In the original equation \((3)^x = 18\), converting it to logarithmic form helps us isolate \(x\), making the solution straightforward. Using the definition, we write it as \(x = \log_3(18)\).
This conversion is a powerful tool in simplifying equations and understanding relationships between quantities.Converting to logarithmic form can make seemingly complex problems easier to handle.

solving equations

Solving equations involves finding an unknown variable that makes the equation true. In exponential equations like \(4(3)^x = 72\), methods such as dividing out constants and using logarithms are frequently used.

Here's how we approach solving this specific equation:

• First, simplify by dividing both sides by 4, giving \((3)^x = 18\).
• Next, express the exponential form in logarithmic terms to find \(x\).

This makes use of the logarithm's ability to transform exponential relationships into more manageable forms.
Solving equations is a fundamental skill in algebra, requiring analytical thinking and a clear understanding of mathematical properties.
Breaking down equations into simpler steps increases accuracy and comprehension.

base conversion

In the context of logarithms, base conversion is the method used to change from one base to another. Often, logarithms are calculated in one base, like 10 or \(e\), even when the original problem uses a different base.
To convert a logarithm from one base to another, use the change of base formula:

• For any logarithm \(\log_b(a)\), it can be rewritten as \(\frac{\log_k(a)}{\log_k(b)}\), where \(k\) is the new base.

In the example discussed, \(x = \log_3(18)\) could be expressed with base 10 (common logarithm) as \(x = \frac{\log(18)}{\log(3)}\).
This conversion allows flexibility in calculating logarithmic values using standard calculators.

A thorough understanding of base conversion is vital, as it often facilitates easier computation and clearer interpretation of logarithmic expressions.