Problem 16 - Calculus and its applications

Question

Solve for \(x\). $$ \log _{4} 1 / 16=x $$

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Rewriting the Logarithmic Equation

To solve \( \log_{4} \frac{1}{16} = x \), first understand that this means finding what power \(x\) that the base 4 must be raised to in order to equal \(\frac{1}{16}\).

Step 2: Expressing 1/16 as a Power of 4

Recognize that \(\frac{1}{16}\) can be rewritten as a power of 4. Knowing that \(16 = 4^2\), we have \(\frac{1}{16} = \frac{1}{4^2} = 4^{-2}\).

Step 3: Equating Powers of 4

Now the equation \( \log_{4} \frac{1}{16} = x \) implies that \(4^x = \frac{1}{16}\). Since \(\frac{1}{16} = 4^{-2}\), we equate this to find: \(4^x = 4^{-2}\).

Step 4: Solving for x

Since the bases are the same (4), we can equate the exponents: \(x = -2\). Thus, the solution is \(x = -2\).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions that help us solve equations involving exponentiation. They are the reverse operation of exponentiation, which means they tell us the power to which a base number must be raised to produce a given number.
For example, in the equation \(\log_{4} \frac{1}{16} = x\), \(4\) is the base, \(\frac{1}{16}\) is the result of the base raised to a power, and \(x\) is the unknown exponent we are trying to find.

Understanding logarithmic functions requires knowing two main things:

• Base: It's the number that gets raised to a power (like \(4\) in our problem).
• Argument: It's the number you want to get when you raise the base to a power (like \(\frac{1}{16}\) here).

With logarithms, you "undo" the exponentiation to find the exponent. This makes them very useful for solving equations where the unknown appears as an exponent.

Exponentiation

Exponentiation is a key mathematical operation and it involves raising a number, called the base, to the power of an exponent. It looks like this: \(b^e\), where \(b\) is the base and \(e\) is the exponent. In practical terms, this means multiplying the base by itself \(e\) times.
In the exercise \(\log_{4} \frac{1}{16} = x\), we are looking for the power \(x\) that makes \(4^x = \frac{1}{16}\).

What you need to know about exponentiation:

• It is a recurring process: The base repeatedly multiplies itself as defined by the exponent.
• When dealing with negative exponents like \(4^{-x} = \frac{1}{4^x}\), they indicate division rather than multiplication.
• Recognizing common powers of a number helps to easily solve equations, as in the example where 16 was recognized as \(4^2\), hence \(\frac{1}{16}\) as \(4^{-2}\).

Understanding exponentiation as repeatedly multiplying or dividing helps make solving equations quicker and clearer.

Base Transformation

Base transformation in logarithms can simplify equations by equating different expressions of the same number. It involves changing how a number is expressed without changing its value. For instance, \( \frac{1}{16} \) can be expressed as \(4^{-2}\) because \(16 = 4^2\).
Recognizing patterns like these is crucial for solving logarithmic equations, as demonstrated in the exercise where we used base 4:

Here's how you can handle base transformation:

• Identify that the same base can express different numbers - like recognizing that \(\frac{1}{16}\) is another way to express \(4^{-2}\).
• This method simplifies equating the logarithm’s argument with a known power of the base.
• By maintaining the same base during transformation, one can solve logarithmic equations more directly, as seen with \(\log_{4} \frac{1}{16} = x\).

Mastering the ability to transform bases flexibly is a big advantage in simplifying and solving logarithmic equations effectively.