Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3$$10^{-3}=0.001\)

Step-by-Step Solution

Answer

The logarithmic form of the equation \(10^{-3}=0.001\) is \(\log_{10}0.001 = -3\).

Step 1: Identify the Elements

In the equation \(10^{-3}=0.001\), '10' is the base (b), '-3' is the exponent (y), and '0.001' is the result (x).

Step 2: Formulate the Logarithmic Equation

Using the logarithmic definition \(b^y = x\) can be written as \(\log_bx = y\), substitute the identified base, exponent and result into the logarithmic form. Therefore, the logarithmic form of \(10^{-3}=0.001\) will be \(\log_{10}0.001 = -3\).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics that denote repeated multiplication of a number by itself. Simply put, when you see a number written with an exponent, such as \(10^{-3}\), the base number is 10, and the exponent is -3. This indicates that 10 is being multiplied by itself with a specific property due to the negative exponent: it involves division.

For example, \(10^{-3}\) is equivalent to \(\frac{1}{10^3}\), which can further be broken down to \(\frac{1}{10 \times 10 \times 10}\). This operation equalizes to \(\frac{1}{1000}\) or 0.001. Here, the negative exponent signals that you're dealing with a fraction instead of a whole number. So, using exponents helps in simplifying multiplication and division repeated over many times.

Key Points about Exponents:

• Positive Exponents: Indicate multiplication of the base.
• Negative Exponents: Indicate division, producing a smaller result.
• Zero Exponent: Any non-zero number with an exponent of zero equals 1.

Base

The term 'base' in mathematics refers to the number that is being multiplied in an exponential expression. In the expression \(10^{-3}\), the base is 10. This means that 10 is the number we are working with when using exponents or logarithms.

The base plays a critical role in determining the growth or decay in exponential expressions. It's essentially the "foundation number" upon which the exponent acts. In different scenarios, the base can be any number, but common bases include numbers like 2, 10, and even the natural base \(e\).

Important Aspects of the Base:

• Logarithmic Base: When converting between logarithmic and exponential forms, the base of the log is the same as the base in the exponent. In our example, the base "10" remains consistent whether in exponential or logarithmic form.
• Impact on Growth: A larger base number increases faster in positive exponents and decreases slower with negative exponents.

Logarithmic Form

Logarithmic form is a way to express equations that employ exponents in a different style that emphasizes the concept of a logarithm. This form allows us to solve for an unknown exponent in expressions and is particularly useful in various scientific fields.

In the original exercise given, you transformed \(10^{-3} = 0.001\) into its logarithmic form \(\log_{10} 0.001 = -3\). In layman's terms, this conversion asks, "To what power must we raise 10 to get 0.001?" The logarithmic form precisely answers that it is raised to the power of -3.

Transitioning to Logarithmic Form:

• Each logarithmic equation follows the structure \(\log_b x = y\) where \(b^y = x\).
• Allows solving for unknowns when direct calculation isn't straightforward.
• Helpful in contexts involving exponential growth or decay, such as in finance and science.

Understanding and using the logarithmic form can thus greatly clarify the roles and relationships of numbers within exponential equations.