Problem 1 - 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) \(\left(g^{-1} \circ f^{-1}\right)(x)\) \(\frac{x+7}{6}\) $$ 2^{7}=128 $$

Step-by-Step Solution

Answer

\(\log_2(128) = 7\)

Step 1: Convert Exponential to Logarithmic Form

The exponential statement given is \(2^7 = 128\). To convert an expression from exponential form \(b^y = x\) to logarithmic form, you use \(\log_b(x) = y\). Here, the base \(b\) is 2, \(y\) is 7, and \(x\) is 128.

Step 2: Write the Logarithmic Expression

Using the conversion rule, rewrite the expression \(2^7 = 128\) in logarithmic form as \(\log_2(128) = 7\). This shows that the logarithm base 2 of 128 is equal to 7.

Key Concepts

Exponential Form

The exponential form is a mathematical way of expressing repeated multiplication of a number by itself. In general terms, it is written as \(b^y = x\), where:

• \(b\) is the base number.
• \(y\) is the exponent or power, indicating how many times the base is used as a factor.
• \(x\) is the result after the base is multiplied by itself \(y\) times.

For example, in the expression \(2^7 = 128\), the base \(b\) is 2, the exponent \(y\) is 7, and the result \(x\) is 128. This means 2 is multiplied by itself 7 times to equal 128. Understanding exponential form is essential for grasping the concept of logarithms and the conversion process between these two mathematical ideas.

Base of a Logarithm

When dealing with logarithms, understanding the base is crucial because it sets the foundation for how a number is expressed in both exponential and logarithmic form. In the context of logarithms, the base is the number that is raised to a power. It is denoted as \(b\) in logarithmic expressions like \(\log_b(x)\). The base is critical for defining the relationship between the exponential and logarithmic forms.For example, in the expression \(\log_2(128) = 7\), the base is 2. This indicates that 2 needs to be raised to the power of 7 to equal 128. The choice of base can vary depending on the context, but common bases include 10 for common logarithms and \(e\) for natural logarithms. Understanding the base helps ease the transition between exponential and logarithmic expressions, aiding in the conversion and comparison of numerical forms.

Conversion Between Forms

Conversion between exponential form and logarithmic form is a fundamental skill in mathematics, especially when working with equations involving growth, decay, or scales. To convert from exponential form \(b^y = x\) to logarithmic form, you rearrange it into \(\log_b(x) = y\).

• Start with the exponential expression. For example, \(2^7 = 128\).
• Identify the base \(b\) (2), the exponent \(y\) (7), and the result \(x\) (128).
• Rearrange it into the logarithmic form: \(\log_2(128) = 7\).

This showcases that 2 must be raised to the power of 7 to yield 128. Understanding this conversion is vital, particularly for solving logarithmic equations or modeling exponential growth and decay scenarios.

Understanding Exponents

Exponents are a shorthand way to represent repeated multiplication of a number by itself, simplifying mathematical expressions and calculations. The exponent, also called the power or index, tells you how many times the base is multiplied by itself.In the expression \(b^y = x\), \(y\) is the exponent which shows how many times the base \(b\) is used as a factor. For example, in \(2^7 = 128\), the exponent is 7, indicating seven iterations of 2 multiplied together. Understanding exponents helps in various mathematical contexts, like simplifying expressions, calculating large numbers, and comprehending logarithms, which inversely use exponents to determine required powers for numbers.