Problem 17 - Intermediate Algebra

Question

For Problems \(11-20\), write each of the following in exponential form. For example, \(\log _{2} 8=3\) becomes \(2^{3}=8\) in exponential form. $$ \log _{2}\left(\frac{1}{16}\right)=-4 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Base

The logarithmic expression is \( \log_{2}\left(\frac{1}{16}\right)=-4 \). Here, the base of the logarithm is \( 2 \). In exponential form, this base becomes the base of the exponent.

Step 2: Identify the Exponent

The logarithmic equation \( \log_{2}\left(\frac{1}{16}\right)=-4 \) indicates that the exponent to which the base \( 2 \) must be raised to obtain \( \frac{1}{16} \) is \( -4 \).

Step 3: Write in Exponential Form

Combine the base and the exponent to express the equation in exponential form: \( 2^{-4} = \frac{1}{16} \). This shows that raising 2 to the power of \(-4\) gives the value \( \frac{1}{16} \).

Key Concepts

Logarithms

Logarithms are essentially the reverse operations of exponents. They help answer questions like "how many times must we multiply the base to get this number?" If you see a logarithm in the form \( \log_b(a) = c \), it simply means that \( b \) raised to the power \( c \) equals \( a \). For example, in the expression \( \log_2(\frac{1}{16}) = -4 \), we're being asked, "to what power must 2 be raised to get \( \frac{1}{16} \)?" The answer here is \(-4\).

Logarithms can seem tricky at first, but they can simplify many calculations in mathematics. They are particularly handy for dealing with very large numbers. Instead of handling big exponents, logarithms allow us to work with their more manageable indices, making those big problems much smaller.

Exponents

Exponents are a way to express repeated multiplication of the same number. When you write \( b^n \), it means you're multiplying the base \( b \) by itself \( n \) times. For instance, \( 2^3 \) equals \( 2 \times 2 \times 2 = 8 \).

In our example, the expression \( 2^{-4} \) signifies that you take the base 2 and multiply by its reciprocal four times, which results in \( \frac{1}{16} \). Negative exponents indicate division or fractions. Specifically, \( b^{-n} \) is the same as \( \frac{1}{b^n} \). This makes it clear why \( 2^{-4} = \frac{1}{16} \): you're essentially dividing 1 by 2 four times.

Exponents simplify calculations involving large numbers or fractions, letting you express them neatly with very little writing.

Base and Exponent

In any exponential expression, it's important to identify two parts: the base and the exponent. The base is the main number being multiplied, while the exponent tells you how many times to multiply the base by itself. For example, in \( 2^3 \), 2 is the base, and 3 is the exponent, implying that you multiply 2 by itself three times.

When converting from logarithmic form to exponential form, as we see in \( \log_2\left(\frac{1}{16}\right) = -4 \), the base of the logarithm (2) becomes the base of the exponent, and the result (-4) becomes the exponent itself.

This understanding makes it easier to interpret and rewrite expressions. Recognizing the base and exponent clarifies the relationship between numbers and how they result in the final outcome of any exponential expression.