Problem 16 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{1 / 4} 16=-2 $$

Step-by-Step Solution

Answer

The exponential form is \(\left(\frac{1}{4}\right)^{-2} = 16\).

Step 1: Understand the Logarithmic Equation

A logarithmic equation is of the form \(\log_b x = y\). This means \(b^y = x\). In the given equation, \(\log_{1/4} 16 = -2\), where the base \(b = \frac{1}{4}\), the number \(x = 16\), and the exponent \(y = -2\).

Step 2: Convert to Exponential Form

Use the formula \(\log_b x = y\) which converts to the exponential form as \(b^y = x\). Applying this to our problem: \(\left(\frac{1}{4}\right)^{-2} = 16\).

Step 3: Simplification

Simplifying \(\left(\frac{1}{4}\right)^{-2}\), we can convert the negative exponent. Recall that \(a^{-n} = \frac{1}{a^n}\), so \(\left(\frac{1}{4}\right)^{-2} = \left(\frac{4}{1}\right)^{2} = 4^{2}\).

Step 4: Verification

Calculate \(4^{2}\) which equals 16, thus verifying the exponential form is correctly derived as \(16 = 16\). This confirms our solution is accurate.

Key Concepts

Logarithmic Equations

Logarithmic equations involve the relationship between the logarithm, its base, and an exponent. Understanding them is key to solving many math problems. A logarithmic equation typically looks like \( \log_b x = y \). Here:

• \(b\) is the base of the logarithm.
• \(x\) is the number we are taking the logarithm of.
• \(y\) is the result or exponent.

Consider the example \( \log_{1/4} 16 = -2 \). This means that raising the base \( \frac{1}{4} \) to the power of \(-2\) equals \(16\). It's crucial to recognize what each part represents in a logarithmic equation so we can accurately convert it into an exponential equation.

Exponents

Exponents serve as a shorthand for repeated multiplication of a base number. The expression \( b^y \) is read as \( b \) raised to the power of \( y \). In our logarithmic equation \( \log_{1/4} 16 = -2 \), the exponent \(-2\) shows how many times the base \( \frac{1}{4} \) should be multiplied by itself to result in \(16\).Here's what you need to know about exponents:

• The exponent tells us how many times to use the base in a multiplication.
• A negative exponent means you take the reciprocal of the base, then apply the positive exponent.

For example, in \( (\frac{1}{4})^{-2} \), this means taking the reciprocal of \( \frac{1}{4} \), which is \(4\), and then squaring it to find \(16\). So, the negative exponent flips our fraction before applying the power.

Converting Forms

Converting between logarithmic and exponential forms is pivotal to understanding these types of mathematical problems. Logarithms and exponents are two sides of the same coin, and converting forms often unravels complex equations.To convert a logarithmic equation \( \log_b x = y \) to an exponential equation, use the relationship \( b^y = x \). This conversion is crucial as it allows further simplification and solving.Here are steps to convert the exercise's logarithmic form to exponential form:

• Identify the base, number, and exponent from the logarithmic equation.
• Rearrange the equation into the form \( b^y = x \).
• Simplify the resulting exponential expression if necessary.

In our example, \( (\frac{1}{4})^{-2} = 16 \) becomes exponential form after identifying the base and rearranging properly. Simplifying gives \( 4^2 = 16 \), which can easily be verified, ensuring precision in understanding and solving the problem.