Problem 10 - Algebra and Trigonometry

Question

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } 10^{3}=1000} & {\text { (b) } 81^{1 / 2}=9}\end{array} $$

Step-by-Step Solution

Answer

(a) \( \log_{10}(1000) = 3 \), (b) \( \log_{81}(9) = \frac{1}{2} \).

Step 1: Identify the Format

An exponential equation can be expressed in logarithmic form. Remember that if you have an exponential equation of the form: \[ a^b = c \] you can express it logarithmically as: \[ ext{log}_a(c) = b \] where \( a \) is the base, \( b \) is the exponent, and \( c \) is the result.

Step 2: Convert First Equation

We are given the first equation as: \[ 10^3 = 1000 \] Using the logarithmic form, it becomes: \[ ext{log}_{10}(1000) = 3 \] This shows that 10 raised to the power of 3 is equal to 1000.

Step 3: Convert Second Equation

The second given equation is: \[ 81^{1/2} = 9 \] Applying the logarithmic conversion, we have: \[ ext{log}_{81}(9) = \frac{1}{2} \] This indicates that 81 raised to the power of 1/2 equals 9.

Key Concepts

Exponential Equations

Exponential equations are a fundamental aspect of mathematics, involving expressions where variables appear as exponents. In an equation like \( 10^3 = 1000 \), "10" is the base and "3" is the exponent. This indicates that the base number, when raised to the exponent, yields the result, "1000." The overall structure of an exponential equation can generally be expressed as \( a^b = c \), where:

• \( a \) is the base
• \( b \) is the exponent
• \( c \) is the result

Exponential equations are pivotal in various scientific and mathematical contexts, helping to describe growth processes, radioactive decay, and many more phenomena. Understanding the components of these equations makes it possible to convert them into other forms, like logarithmic form.

Base and Exponent

In the world of exponents, the base and the exponent are vital components that form exponential equations. The base is the number being multiplied by itself, while the exponent indicates how many times the base is used as a factor. For example, in the equation \( 81^{1/2} = 9 \):

• The base is 81
• The exponent is \( \frac{1}{2} \)
• The result is 9

This setup allows for easy calculation of powers and roots. A key part of understanding exponents involves recognizing patterns, such as when the exponent is a fraction (like \( \frac{1}{2} \)), which represents a square root. Thus, \( 81^{1/2} \) is indeed the square root of 81, which is 9. Grasping these concepts helps in comprehending more complex mathematics, including the transition to logarithmic expressions.

Logarithm Rules

Understanding logarithms is essential for working with exponential expressions. A logarithm answers the question, "To what power must the base be raised to produce this number?" If you see a logarithmic expression like \( \log_{10}(1000) = 3 \), it tells you that 10 raised to the power of 3 gives you 1000. Some fundamental rules of logarithms include:

• \( \log_a(1) = 0 \): Any base raised to the power of zero is one.
• \( \log_a(a) = 1 \): Any number raised to the power of one gives the number itself.
• \( \log_a(a^b) = b \): Indicates the exponentiation is the inverse of the logarithm.

These rules make it simpler to manipulate and understand expressions in both logarithmic and exponential forms. Mastery of these rules is crucial when solving logarithmic equations and converting them back to exponential form.

Conversion Between Forms

Conversion between exponential and logarithmic forms is a technique that can simplify complex equations, particularly in areas like calculus and algebra. Knowing how to switch between these forms is significant for ease of solving and comprehension. To convert an exponential equation \( a^b = c \) to logarithmic form, you write: \( \log_a(c) = b \). This step transforms the understanding of the equation into a question: "What power do I need to raise \( a \) to, to get \( c \)?"For instance, with \( 10^3 = 1000 \), converting forms helps to readily see that \( \log_{10}(1000) = 3 \). Similarly, for \( 81^{1/2} = 9 \), it translates into \( \log_{81}(9) = \frac{1}{2} \).By grasping these concepts, you enhance your ability to analyze and solve equations adeptly, making it simpler to understand both mathematical theories and real-world applications. Effective conversion can unlock clarity in otherwise complex mathematical scenarios.