Problem 4 - Algebra for College Students

Question

Write each exponential statement in logarithmic form. For example, \(2^{5}=32\) becomes \(\log _{2} 32=\) 5 in logarithmic form. (c) \(\left(g^{-1} \circ f^{-1}\right)(x)\) \(\frac{x+7}{6}\) $$ 2^{6}=64 $$

Step-by-Step Solution

Answer

\(\log_2 64 = 6\)

Step 1: Understanding the Exponential Statement

We start with the exponential statement given in the problem, which is \(2^6 = 64\). In exponential form, it reads as base 2 raised to the power of 6 equals 64.

Step 2: Identifying Logarithmic Components

In the logarithmic form, every exponential equation \(a^b = c\) can be rewritten as \(\log_a c = b\). Here, the base \(a\) is 2, the result \(c\) is 64, and the exponent \(b\) is 6.

Step 3: Writing the Logarithmic Form

Convert the exponential equation \(2^6 = 64\) into its logarithmic form using the identified components. The logarithmic form is \(\log_2 64 = 6\). This states that the logarithm of 64 with base 2 is 6.

Key Concepts

Exponential Statement

An exponential statement is a fundamental concept in mathematics, where a number, called the base, is raised to the power of another number, known as the exponent. In general terms, an exponential statement is written as \(a^b = c\). Here, \(a\) is the base, \(b\) is the exponent, and \(c\) is the outcome of raising the base to the exponent.
Imagine it as repeatedly multiplying the base number by itself. For example, if you have \(2^6\), this means you multiply 2 by itself 6 times, resulting in 64.
A few key characteristics of exponential statements include:

• The result is always determined once you know the base and the exponent.
• They describe how quantities can increase or decrease exponentially, which is common in natural phenomena, like population growth or radioactive decay.

Transforming an exponential statement into a logarithmic form allows us to reveal the exponent when given the base and the outcome.

Base

The base in an exponential statement is the number that is repeatedly multiplied by itself, as determined by the exponent. In the equation \(2^6 = 64\), the base is 2. This is the number which forms the foundation of our exponential calculation.
Understanding the base is crucial because:

• It determines the 'scale' or 'rate' of growth described by the exponential statement.
• When converting exponential statements to logarithmic form, the base becomes the base of the logarithm, maintaining its identity.

For example, in the logarithmic form \(\log_2 64 = 6\), the base remains at 2, showing how it transfers from one form to the other. The consistent base across different mathematical forms helps in reversing operations, such as moving from logarithmic back to exponential calculations.

Exponent

The exponent in an exponential expression \(a^b = c\) indicates how many times the base \(a\) is multiplied by itself. It essentially provides the power to which the base is raised.
In our example \(2^6 = 64\), the exponent is 6, meaning the base 2 multiplies itself 6 times to reach 64. This is a crucial concept in mathematics as it allows for the compact expression of very large or very small numbers.
The importance of understanding exponents includes:

• They determine the intensity or magnitude of change in exponential growth or decay.
• In logarithmic form, the exponent transitions to the solution of the logarithmic equation, like in \(\log_2 64 = 6\), illustrating how the logarithm uncovers the exponent from the exponential statement.

Thus, comprehending the role of the exponent aids in navigating between different mathematical representations like exponential and logarithmic forms.