Problem 13 - Precalculus

Question

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\log (0.1)=-1\)

Step-by-Step Solution

Answer

The logarithmic equation \( \log (0.1) = -1 \) is rewritten as the exponential equation \( 10^{-1} = 0.1 \).

Step 1: Identify the Base of the Logarithm

In the equation \( \log (0.1) = -1 \), the base of the logarithm is 10. When no base is specified, it is understood to be base 10.

Step 2: Rewrite as Exponential Equation

Using the property \( b^a = c \) if and only if \( \log_b (c) = a \), we rewrite the equation \( \log_{10} (0.1) = -1 \) into its exponential form. According to the theorem, this corresponds to \( 10^{-1} = 0.1 \).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear in exponents. They usually have a form like \( b^x = c \) where \( b \) is the base, \( x \) is the exponent, and \( c \) is the result when \( b \) is raised to the power of \( x \). Exponential equations are fundamental in fields like finance, population dynamics, and physics when modeling growth or decay:

• In finance, they model compound interest, where investment grows at a consistent rate.
• In physics, they describe radioactive decay or cooling processes, showing how quantities decrease over time.
• In population studies, they predict population growth based on a constant growth rate.

When solving exponential equations, converting them to logarithmic form can simplify the process, making it easier to isolate the variable in the exponent.

Theorem 6.2

Theorem 6.2 is a mathematical bridge between exponential and logarithmic functions. It states that: \[b^a = c \text{ if and only if } \log_b(c) = a\]This theorem allows us to transition between an exponential equation and its corresponding logarithmic form, providing a versatile tool for solving equations: * When you have \( b^a = c \), you can find the logarithmic representation \( \log_b(c) = a \), giving clarity to what the exponent \( a \) represents.* Conversely, knowing \( \log_b(c) = a \) allows you to express it as \( b^a = c \), understanding how the base raised to \( a \) yields \( c \). Using Theorem 6.2, as seen in the given exercise, eases the manipulation of equations between their logarithmic and exponential forms, aiding in their analysis and solution.

Base 10 Logarithm

The base 10 logarithm is commonly referred to as the 'common logarithm,' used extensively in scientific and practical computations. Its notation is either \( \log(c) \) or \( \log_{10}(c) \), with 10 as its base. Base 10 logarithms are intuitive when dealing with numbers in decimal systems:

• Calculations involving powers of 10 become easier to interpret, as each logarithmic step represents a power of 10.
• The base 10 logarithm of a number reflects how many times 10 must be multiplied by itself to reach that number.

In the exercise, the equation \( \log(0.1) = -1 \) utilizes base 10, implying that \( 10^{-1} = 0.1 \). This usage underscores the base 10 logarithm's role in simplifying expressions and translating them into exponential form, especially in educational contexts where decimal concepts are foundational.