Problem 10 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the logarithmic equation in exponential form. For example, the exponential form of \(\log _{5} 25=2\) is \(5^{2}=25.\) $$\log _{10} \frac{1}{1000}=-3$$

Step-by-Step Solution

Answer

The exponential form of \(\log _{10} \frac{1}{1000}=-3\) is \(10^{-3}=\frac{1}{1000}\).

Step 1: Understand the Logarithmic Equation

In the equation \(\log _{10} \frac{1}{1000}=-3\), '10' is the base of the logarithm, '1/1000' is the argument of the logarithm and '-3' is the value of the logarithm. The aim is to convert this into an equivalent exponential form.

Step 2: Conversion to Exponential Form

To convert the logarithm to exponential form, think of it in these terms: the base (10) raised to the value of the logarithm (-3) equals the argument of the logarithm (1/1000). This gives us the equivalent exponential form.

Step 3: Write Exponential Form

The exponential form of \(\log _{10} \frac{1}{1000}=-3\) is thus \(10^{-3}=\frac{1}{1000}\).

Key Concepts

Logarithmic Form

The logarithmic form is a way to express a mathematical relationship where you are trying to find the power to which a base number must be raised to get a certain value. It's presented as \( \log_b a = c \), where \( b \) is the base, \( a \) is the argument, and \( c \) is the logarithm value. For example, in the equation \( \log_{10} \frac{1}{1000} = -3 \), the log base 10 of \( \frac{1}{1000} \) is \(-3\).

This equation tells us that if we start with a base of 10, we need to raise it to the power of \(-3\) to get \( \frac{1}{1000} \).

Identifying the parts:

• Base \( (b) \): The number 10 in this equation.
• Argument \( (a) \): The fraction \( \frac{1}{1000} \), representing the result.
• Logarithm Value \( (c) \): The number \(-3\), indicating the exponent.

Hence, the logarithmic form is simply a way to represent these elements and their relationship.

Exponential Form

The exponential form is another way to present the same relationship you see in a logarithmic equation. It 'undoes' the logarithm to show how the base is exponentiated to reach the argument. The general representation is \( b^c = a \).

In the example of \( \log_{10} \frac{1}{1000} = -3 \), converting this into exponential form involves raising the base 10 to the exponent \(-3\):

\[10^{-3} = \frac{1}{1000}\]

The components in the exponential equation consist of:

• Base \( (b) \): The number 10, which is being raised to power.
• Exponent \( (c) \): The power, here it's \(-3\), adjusting the base for the result.
• Result \( (a) \): The outcome is \( \frac{1}{1000} \).

It essentially breaks down to using an exponent \(-3\) to calculate or verify the result \( \frac{1}{1000} \) from a base of 10.

Base of Logarithm

Understanding the base of logarithms is crucial because it determines the system of growth or decay in the problem. The base in a logarithmic function (and its corresponding exponential form) is denoted as \( b \). It is essentially the number that is being repeatedly multiplied in the exponential form.

In our example, \( \log_{10} \frac{1}{1000} = -3 \), the base is 10. This specifies that we are operating in a decimal system, as 10 is a common base used in scientific calculations and logarithmic computations.

Key points about the base:

• The base (10 in this instance) tells you which number is being used in repeated multiplication or division.
• The logarithm with this base becomes more easily interpreted as simply the power the base must be raised to obtain a particular number.
• The choice of base can influence how exponents and results are represented and interpreted.

In general, the base is fundamental to both the logarithmic form and exponential form, linking them through this consistent multiplier.