Problem 3 - A Graphical Approach to Precalculus with Limits

Question

For each statement, write an equivalent statement in logarithmic form. Do not use a calculator. $$\left(\frac{1}{2}\right)^{-4}=16$$

Step-by-Step Solution

Answer

Equivalent logarithmic form is \( \log_{\frac{1}{2}}(16) = -4 \).

Step 1: Understand the Exponential Form

The given statement in exponential form is \( \left(\frac{1}{2}\right)^{-4}=16 \). This means we have a base \( \frac{1}{2} \), an exponent \( -4 \), and the result is \( 16 \).

Step 2: Recall the Logarithmic Form

The general form to convert an exponential statement \( b^x = y \) to logarithmic form is \( \log_b(y) = x \). Here, \( b \) is the base, \( x \) is the exponent, and \( y \) is the result.

Step 3: Apply the Conversion

Using the form \( \log_b(y) = x \), identify \( b = \frac{1}{2} \), \( y = 16 \), and \( x = -4 \). Substituting these values into the logarithmic form gives us \( \log_{\frac{1}{2}}(16) = -4 \).

Key Concepts

Exponential Form

The exponential form is a way to write numbers that have been raised to a power, where this structure is expressed as \( b^x = y \). Here, \( b \) represents the base, \( x \) is the exponent, and \( y \) is the result. For example, in the statement \( \left(\frac{1}{2}\right)^{-4} = 16 \), \( \frac{1}{2} \) is the base, \( -4 \) is the exponent, and 16 is the result.
Understanding exponential form is crucial because it allows you to express large numbers compactly or to break down complex calculations. The base tells you the number that is multiplied by itself, whereas the exponent tells you how many times the base is used as a factor. In this context, having a negative exponent like \(-4\) means you are taking the reciprocal of the base raised to the positive exponent.
This form is very useful in various math and science disciplines to describe exponential growth, decay, and other phenomena in a simplified manner.

Base and Exponent

In mathematical expressions involving exponents, the base and the exponent play critical roles. The base is the number that is multiplied by itself, and the exponent tells us how many times to perform this multiplication. Let's take a closer look using \( \left(\frac{1}{2}\right)^{-4} \) as an example:

• Base: The base here is \( \frac{1}{2} \). It's the number you start with before applying the power of the exponent.
• Exponent: The exponent is \(-4\). Because it is negative, it means you take the reciprocal of the base raised to the opposite positive power. So, \( \left(\frac{1}{2}\right)^{-4} \) becomes \( 2^4 \).

These concepts are fundamental in mathematics as they are used in countless applications—from computing compound interest to calculating scientific phenomena. Understanding how bases and exponents interact is key to mastering calculations with powers.

Conversion Between Logarithms and Exponents

Converting between exponential and logarithmic forms is an essential skill in mathematics. It can often simplify solving equations and understanding relationships between numbers. The conversion makes use of the fact that logarithms can be thought of as the inverse operation of exponentiation.

For any exponential expression \( b^x = y \), you can convert this to a logarithmic expression using the formula \( \log_b(y) = x \). The base \( b \) stays the same, \( y \) becomes the number you find the log of, and \( x \) is the result of the logarithm. In our context, with the exponential form \( \left(\frac{1}{2}\right)^{-4} = 16 \), you can easily transform this to the logarithmic form as \( \log_{\frac{1}{2}}(16) = -4 \).

• This transformation shows you that the logarithm gives the exponent to which the base must be raised to get the given result.
• With this understanding, operations like solving for unknown exponents or bases become more intuitive.

Mastering the conversion between these two forms is a strong asset for anyone looking to deepen their mathematical literacy.