Problem 9 - Algebra and Trigonometry

Question

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } 5^{3}=125} & {\text { (b) } 10^{-4}=0.0001}\end{array} $$

Step-by-Step Solution

Answer

(a) \(\log_5(125) = 3\); (b) \(\log_{10}(0.0001) = -4\).

Step 1: Understand Exponential to Logarithmic Conversion

Understanding that an exponential equation of the form \(a^x = b\) can be expressed in logarithmic form as \(\log_a(b) = x\). This conversion involves identifying the base \(a\), the exponent \(x\), and the result \(b\).

Step 2: Convert Part (a) Equation

The given equation is \(5^3 = 125\). Here, the base \(a = 5\), the exponent \(x = 3\), and the result \(b = 125\). According to the conversion rule, this equation in logarithmic form is \(\log_5(125) = 3\).

Step 3: Convert Part (b) Equation

The given equation is \(10^{-4} = 0.0001\). Here, the base \(a = 10\), the exponent \(x = -4\), and the result \(b = 0.0001\). Using the conversion rule, this equation in logarithmic form is \(\log_{10}(0.0001) = -4\).

Key Concepts

Exponential to Logarithmic Conversion

Converting from exponential to logarithmic form is a fundamental concept in mathematics that helps to reframe how we think about numbers and powers. An exponential form is typically written as \(a^x = b\), where \(a\) is the base, \(x\) is the exponent, and \(b\) is the result.
To convert this relationship into a logarithmic form, you use the format \(\log_a(b) = x\). Here, \(\log\) signifies that you are looking for an exponent. Essentially, you are asking, "To what power must I raise \(a\) to get \(b\)?" This transformation is particularly useful as it allows for simplification of complex calculations, especially when dealing with large numbers. It's also the basis for many applications in science and engineering.

Base and Exponent

In both the exponential and logarithmic forms, understanding the base and exponent is crucial for proper conversion and interpretation.

• Base \((a)\): This is the number that is multiplied by itself a certain number of times. In our example, for \(5^3 = 125\), the base is 5.
• Exponent \((x)\): This shows how many times the base is multiplied by itself. In the example, 3 is the exponent, indicating that 5 is used as a factor 3 times.

In logarithmic terms, the base () is very important as it defines the scale of growth or decay. A change in base changes the entire meaning of the equation. Thus, converting between these forms requires careful identification of the base and exponent to ensure accuracy.

Logarithmic Equations

Logarithmic equations are equations that involve logarithms with variables in the base or other parts of the logarithm. These equations are solved by converting to exponential form, which often simplifies solving.
For example, the logarithmic equation \(\log_5(125) = 3\) states, "The exponent needed to raise 5 to yield the number 125 is 3." Such an equation can help in solving for unknowns either in the base or the result in more complex equations.

• Logarithmic equations are often used to model real-world phenomena such as growth, decay, and scale transformation due to their nature of showing relationships between multi-scale events.
• Solving these equations usually involves applying properties like the product, quotient, or power rules of logarithms.

Mathematical Transformations

Mathematical transformations involve changing the form of an equation to make it easier to work with or to solve.
The conversion from exponential to logarithmic form is a classic example of transformation. It's a tool that enables simplification and opens up new ways of problem-solving, making it easier to grasp complex relationships.
For example, the transformation of \(10^{-4} = 0.0001\) into the logarithmic form \(\log_{10}(0.0001) = -4\) changes perspective but retains equality. It allows us to use the intuitive properties of logarithms for computational simplicity in algebra and calculus. Transformations also help to graph or visualize equations by providing a bridge between different mathematical representations.