Problem 19 - 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. $$ \log _{10} 0.001=-3 $$

Step-by-Step Solution

Answer

The exponential form is \(10^{-3} = 0.001\).

Step 1: Identify Components of Logarithmic Expression

The given logarithmic expression is \(\log _{10} 0.001 = -3\). From this expression, we identify the base as 10, the argument as 0.001, and the result of the logarithm as -3.

Step 2: Understand Exponential Form

In exponential form, a logarithmic equation \(\log_b a = c\) is rewritten as \(b^c = a\). This means the base raised to the power of the logarithm's result equals the argument.

Step 3: Convert to Exponential Form

Using the logarithmic expression \(\log _{10} 0.001 = -3\), we can write this in exponential form by stating that the base 10 raised to the power of -3 equals the argument 0.001. Thus, the exponential form is \(10^{-3} = 0.001\).

Key Concepts

Logarithmic Expression

Logarithmic expressions are essential in understanding the relationship between numbers in mathematics. Here, a logarithm is essentially the inverse operation of exponentiation. To make sense of this, consider the logarithmic expression \(\log_b a = c\). This tells us a couple of things:

• The base \(b\) is determined.
• The argument \(a\) is what the base \(b\) is being raised to produce.
• The result \(c\) is the power the base \(b\) must be raised to in order to obtain \(a\).

Logarithms help simplify problems dealing with multiplication and division by transforming them into addition and subtraction.

Base and Exponent

The base and exponent are two key components of exponential expressions. The base, usually denoted as \(b\), is the number that gets multiplied by itself. The exponent \(c\), tells us how many times the base is used as a factor. For instance, in the expression \(b^c = a\), the essence of this notation is:

• \(b\) is the base and determines the fundamental unit of multiplication.
• \(c\) is the exponent and indicates how often the base is multiplied by itself.

When we see an expression like \(10^{-3} = 0.001\), it communicates that the base, 10, is being raised to a power of -3, resulting in a smaller number because of the negative exponent. Raising a number to a negative power essentially finds the reciprocal of that number to the positive power.

Converting Logarithms to Exponentials

Converting logarithmic expressions into exponential form is a straightforward but important skill. This process involves restructuring a logarithmic equation \(\log_b a = c\) as an exponential equation \(b^c = a\). Let's break it down:

• Identify the logarithm's base, which becomes the base in the exponential form.
• The result of the logarithm becomes the exponent in the exponential form.
• The argument of the logarithm is the result (or what you want to get) in the exponential equation.

For example, the expression \(\log_{10} 0.001 = -3\) translates to \(10^{-3} = 0.001\). In this instance, the base is 10, the exponent is -3, and the argument (what you get) is 0.001. Practicing this conversion skill reinforces understanding of both logarithms and exponentials, providing a flexible approach to solving various mathematical problems.