Problem 31 - Elementary and 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 form is \( 10^{-1} = 0.1 \).

Step 1: Understanding the Logarithmic Equation

The given logarithmic equation is \( \log 0.1 = -1 \). In logarithmic form, \( \log_b a = c \) means \( b^c = a \). Here, our base \( b \) is assumed to be 10 by default since no base is specified in common logarithms.

Step 2: Rewriting in Exponential Form

To convert \( \log 0.1 = -1 \) to its exponential form, we use the relationship \( b^c = a \). Using base 10, we have \( 10^{-1} = 0.1 \).

Step 3: Expressing the Final Answer

The exponential form of the logarithmic equation \( \log 0.1 = -1 \) is \( 10^{-1} = 0.1 \). This correctly represents the same relationship between the base, exponent, and result.

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve logarithms. They are often used to solve problems where a number is raised to an unknown power. Logarithmic equations have the form \( \log_b a = c \), meaning that \( b^c = a \), where:

• b denotes the base of the logarithm.
• a is the number we are applying the logarithm to.
• c is the exponent (or power) that the base is raised to in order to equate to a.

Logarithmic equations simplify complex calculations, especially in exponential growth or decay scenarios. They help in understanding the relationship between exponential growth and its inverse exponential functions. When solving, make sure to consider the base of the logarithm, as it can change the form and result.
Overall, logarithmic equations are a fundamental part of algebra and precalculus that facilitate problem solving where direct calculations are cumbersome.

Base 10 Logarithms

Base 10 logarithms, often referred to as common logarithms, use 10 as their base. This means when you see \( \log a \) without a base, it typically means \( \log_{10} a \).

• They are widely used in many scientific and engineering fields due to their compatibility with our decimal number system.
• This type of logarithm simplifies a number by showing how many times you need to multiply 10 to get that number.

For example, \( \log 100 \) is 2 because \( 10^2 = 100 \). In daily applications, base 10 logarithms make it easier to handle large and complex numbers by converting them into manageable values.
It's essential to recognize that when no base is specified for a logarithm, the base is presumed to be 10. This convention comes in handy when converting between forms or simplifying equations.

Converting Forms

Converting between logarithmic and exponential forms is a key skill in algebra. This involves moving from \( \log_b a = c \) to an equivalent expression \( b^c = a \), or vice versa.

• Start by identifying the components: the base \( b \), the argument \( a \), and the exponent \( c \).
• To convert a logarithmic equation into an exponential one, express the relationship in terms of powers of \( b \).

For example, converting \( \log 0.1 = -1 \) to its exponential form involves recognizing that the base is 10. Thus, in exponential form it becomes \( 10^{-1} = 0.1 \).
Why Convert?

• Different forms can make solving or understanding particular problems easier.
• Exponential equations can be more intuitive for solving real-world growth or decay problems.

Remember that converting between these forms can aid in visualizing and solving several types of math problems efficiently.