Problem 15 - A Graphical Approach to College Algebra

Question

For each statement, write an equivalent statement in exponential form. $$\log _{10} 0.001=-3$$

Step-by-Step Solution

Answer

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

Step 1: Understand Logarithmic Notation

The expression \(log_{10} 0.001 = -3\) means that 10 raised to what power gives 0.001. Here, the base of the logarithm is 10, and the argument is 0.001.

Step 2: Apply the Definition of Logarithms

The logarithmic expression \(\log_{b}(a) = c\) can be rewritten in its equivalent exponential form as \(b^c = a\). Here, \(b = 10\), \(c = -3\), and \(a = 0.001\).

Step 3: Rewrite in Exponential Form

Using the formula from Step 2, rewrite the logarithmic equation \(\log_{10} 0.001 = -3\) in exponential form as \(10^{-3} = 0.001\). This shows that raising 10 to the power of -3 results in 0.001.

Key Concepts

Logarithmic Notation

Logarithmic notation is a way to express the relationship between exponents and their results. In the expression \(\log_{10} 0.001 = -3\), the notation itself communicates a lot. It means "the power to which 10 must be raised to result in 0.001 is -3." This is a compact way to express what would otherwise be an involved mathematical calculation. By understanding logarithmic notation, you're opening a door to simplifying the way we handle large or small numbers.

• "log" represents the logarithm.
• The subscript "10" is the base of the logarithm.
• The main number, "0.001," is the argument or the result you're looking for.
• The "-3" is the exponent, answering the question "Power to what?"

Logarithmic notation is like a language that makes these comparisons more efficient and understandable. It transforms multiplicative relationships into additive ones by focusing on the exponent.

Logarithm Base

The base in a logarithm is one of its most crucial parts and fundamentally influences the calculations. In our example \(\log_{10} 0.001 = -3\), the base is 10. The base tells us what number is being repeatedly multiplied by itself. In simpler terms, it answers what number needs to be raised to a particular power.

Common bases include

• Base 10, which is often used in scientific calculations and is called the "common logarithm." It simplifies multiplying large numbers by using powers of ten.
• Base \(e\), fundamental in calculus and known as the "natural logarithm."
• Base 2, frequently used in computer science, representing binary code.

Understanding the base gives you a foundation for converting logarithmic notation into an exponential expression, as we shall see in the next section.

Exponential Expression

An exponential expression uses a base and an exponent to describe a number. When you see something like \(10^{-3} = 0.001\), it's showing how many times you multiply 10 by itself to reach 0.001. This is what the logarithmic notation was expressing in shorthand.

The conversion from logarithmic to exponential form is straightforward once you have identified:

• The base \(b\), which is 10 in this case.
• The exponent \(c\), which is the -3.
• The result \(a\), which the base raised to the exponent should equal, here being 0.001.

Rewriting the expression from logarithmic to exponential helps visualize the operation as multiplication, offering a clearer perspective of how numbers relate and interact. This simplified view allows for better understanding and manipulation of equations.