Problem 22 - Calculus and its applications

Question

Write an equivalent logarithmic equation. $$ 10^{-1}=0.1 $$

Step-by-Step Solution

Answer

The equivalent logarithmic equation is \(\log_{10}(0.1) = -1\).

Step 1: Understanding the Problem

We need to convert the given equation, which is in the form of an exponential equation, into its equivalent logarithmic form. The equation is given as \(10^{-1} = 0.1\).

Step 2: Identify the Base, Exponent, and Result

In the equation \(10^{-1} = 0.1\), the base is 10, the exponent is -1, and the result is 0.1. These elements are essential for writing the equivalent logarithmic expression.

Step 3: Convert to Logarithmic Form

The general logarithmic form corresponding to the exponential expression \(b^x = a\) is \(\log_b(a) = x\). Here, applying this knowledge, we have the base \(b = 10\), result \(a = 0.1\), and exponent \(x = -1\). Thus, the logarithmic form is \(\log_{10}(0.1) = -1\).

Key Concepts

Exponential Expressions

Exponential expressions are equations that represent numbers in terms of a base raised to a certain power. In the exercise, the exponential expression given is \(10^{-1} = 0.1\). This means that 10 is raised to the power of -1 to get the result 0.1.
Exponential expressions are a powerful way to express repeated multiplication. Here's how they work:

• The base is the number that is being multiplied.
• The exponent tells us how many times to multiply the base by itself.

When the exponent is a negative integer, it means dividing 1 by the base raised to the positive of that exponent. Thus, \(10^{-1}\) translates to dividing 1 by 10, resulting in 0.1. Grasping how exponential expressions work is essential as it is a foundation for converting to logarithmic form.

Logarithmic Form

Converting exponential expressions to logarithmic form is a critical mathematical skill, especially in understanding equations. The exponential form \(b^x = a\) can be rewritten in logarithmic form as \(\log_b(a) = x\). This equation allows us to determine the exponent by understanding the relationship between the numbers involved.
In our example, \(10^{-1} = 0.1\) is converted to \(\log_{10}(0.1) = -1\). Here's why this conversion is useful:

• The logarithmic form provides a different perspective, concentrating on the power to which the base must be raised to result in a certain number.
• It is especially valuable in solving equations where you need to find unknown exponents.

Logarithms are the inverses of exponentials, meaning they can reverse the process of taking powers, making them invaluable for many mathematical computations.

Base and Exponent Identification

Recognition of the base and exponent within a mathematical expression is key to understanding both exponential and logarithmic forms. In exponential expressions like \(10^{-1} = 0.1\), one must correctly identify:

• Base: The constant factor multiplied by itself, here it is 10.
• Exponent: The number indicating how many times the base is used in the multiplication, which in our case is -1.

Correctly identifying these elements is crucial for successfully converting to a logarithmic form, where:

• The base remains the same in the log form (\(b = 10\)).
• The number for which we want the log becomes the result part of the exponential equation (\(a = 0.1\)).
• The exponent from the exponential form becomes the resultant of the logarithmic expression (\(x = -1\)).

Being precise with these identifications allows one to smoothly transition between exponential and logarithmic representations, which is essential for problem-solving.