Problem 14 - Algebra and Trigonometry

Question

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: Understanding the Exponential Form

An exponential equation is typically in the form of \( b^c = a \) where \( b \) is the base, \( c \) is the exponent, and \( a \) is the result. The task here is to re-write these equations in logarithmic form.

Step 2: Converting to Logarithmic Form (a)

The given equation is \( 10^3 = 1000 \). In logarithmic form, this becomes \( \log_{10}(1000) = 3 \), which means the power to which 10 must be raised to get 1000 is 3.

Step 3: Converting to Logarithmic Form (b)

The given equation is \( 81^{1/2} = 9 \). In logarithmic form, this becomes \( \log_{81}(9) = \frac{1}{2} \), indicating that the power to which 81 must be raised to get 9 is 1/2.

Key Concepts

Exponential Equations

Exponential equations are expressions where a number called the base is raised to a power, represented as the exponent. This is a common way to express growth, decay, and other mathematical phenomena. The general structure of an exponential equation is \( b^c = a \), where:

• \( b \) is the base, representing the number being multiplied repeated times.
• \( c \) is the exponent, indicating how many times the base is multiplied by itself.
• \( a \) is the result of the base raised to the exponent.

This equation is fundamental in various scientific and financial calculations, as it allows you to describe exponential growth like population increase, radioactive decay, or compound interest.

Logarithms

Logarithms are the inverse operation to exponentiation. If you think of multiplication and division as opposites, think of exponentials and logarithms as similar opposites. They are used to solve for the exponent in an exponential equation. To express an exponential equation like \( b^c = a \) in logarithmic form, it is written as \( \log_b(a) = c \). Here’s what each part means:

• \( \log_b \) represents the logarithm to the base \( b \).
• \( a \) is the result you know beforehand.
• \( c \) is the unknown exponent we are trying to find.

This operation helps simplify complex calculations, especially in large datasets or when dealing with very large or small numbers. For instance, \( \log_{10}(1000) \) asks, "To what power must 10 be raised to get 1000?" The answer is 3, as in the given equation.

Base and Exponent Relationship

The relationship between the base and the exponent is key to understanding both exponential equations and logarithms. The base is the repeated factor in multiplication, while the exponent tells you how many times to use this base as a factor.For example, in the equation \( 10^3 = 1000 \):

• The base \( 10 \) is multiplied by itself 3 times to produce 1000.
• We can test this as: \( 10 \times 10 \times 10 = 1000 \).

Similarly, in the fractional exponent \( 81^{1/2} = 9 \), the exponent \( \frac{1}{2} \) indicates a square root, which is the same as raising 81 to the power of 0.5. Understanding how bases and exponents interact helps make sense of logarithmic transformations and simplifies our calculations.