Problem 31 - Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log 0.1=-1 $$

Step-by-Step Solution

Answer

The exponential equation is \( 10^{-1} = 0.1 \).

Step 1: Identify Logarithmic Form

The given equation is \( \log 0.1 = -1 \). This is in the logarithmic form of \( \log_b a = c \), where \( a = 0.1 \), and \( c = -1 \).

Step 2: Recall Exponential Form

The exponential form of the logarithmic equation \( \log_b a = c \) is \( b^c = a \). Our task is to identify the base \( b \).

Step 3: Identify the Base

In common logarithms, where the base is not shown, it is implicitly \( b = 10 \). Thus, the equation \( \log 0.1 = -1 \) has a base \( b = 10 \).

Step 4: Convert to Exponential Form

Using the base 10 identified, convert the logarithmic equation to its exponential form. According to \( b^c = a \), we have \( 10^{-1} = 0.1 \). Therefore, the exponential form of the equation is \( 10^{-1} = 0.1 \).

Key Concepts

Exponential Form

Understanding the exponential form is key to converting logarithmic equations. The exponential form of an equation shows us how many times a number, known as the base, needs to be multiplied by itself to reach another number, called the result. This is written as \( b^c = a \). In this equation:

• \( b \) represents the base.
• \( c \) is the exponent, showing the number of times the base is multiplied by itself.
• \( a \) is the result of the expression.

For instance, if we take the equation \( 10^{-1} = 0.1 \), it tells us that the base 10 needs to be used as a factor \(-1\) times, which actually represents the reciprocal of 10, giving us 0.1. This form is very intuitive when you want to understand growth, decay, or simply the conversion from a logarithmic equation.

Logarithmic Form

The logarithmic form of an equation is essentially the reverse of the exponential form. It indicates the power to which the base must be raised to get a certain number. The logarithmic form is written as \( \log_b a = c \). In this notation:

• \( b \) is the base, which must be raised to a certain power.
• \( a \) is the result that \( b \) is being raised to achieve.
• \( c \) represents the power or exponent.

Using the original exercise, \( \log 0.1 = -1 \), means we are looking for the power to which the base 10 must be raised to yield 0.1. The critical realization here is that when no base is specified, it is implicitly \( b = 10 \), making daily tasks, like simple calculations or understanding measurements, much simpler.

Base 10

The concept of base 10, also known as the common logarithm, is a fundamental principle in both mathematics and everyday life. The base 10 system is the default in the absence of an explicitly stated base. This is because humans generally use a decimal system, which simplifies many mathematical operations.Here's why base 10 is special:

• It is used in most scientific calculators and computers as the default logarithmic base.
• The base 10 logarithm of a number explains how many zeros are behind the number, providing a quick way to gauge magnitude.
• Common logarithms help in measuring the intensity of earthquakes, sound levels, and in diverse mathematical fields.

In practical application, understanding that \( \log 0.1 = -1 \) means that 10 raised to get 0.1 requires a power of \(-1\) as seen by the formula \( 10^{-1} = 0.1 \). This simplicity and consistency make the base 10 system highly efficient for daily use and comprehension.