Problem 11 - Algebra and Trigonometry

Question

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{lll}{\text { (a) } 8^{-1}=\frac{1}{8}} & {\text { (b) } 2^{-3}=\frac{1}{8}}\end{array} $$

Step-by-Step Solution

Answer

(a) \( \log_8 \frac{1}{8} = -1 \); (b) \( \log_2 \frac{1}{8} = -3 \).

Step 1: Identify the given equation (a)

The equation provided in part (a) is expressed in exponential form: \( 8^{-1} = \frac{1}{8} \).

Step 2: Rewrite equation (a) in logarithmic form

To rewrite an exponential equation in logarithmic form, use the relationship that if \( a^b = c \), then \( \log_a c = b \). Hence, for the equation \( 8^{-1} = \frac{1}{8} \), it can be rewritten as: \( \log_8 \frac{1}{8} = -1 \).

Step 3: Identify the given equation (b)

The equation provided in part (b) is expressed in exponential form: \( 2^{-3} = \frac{1}{8} \).

Step 4: Rewrite equation (b) in logarithmic form

Using the same transformation as before, if \( a^b = c \), then \( \log_a c = b \). For the equation \( 2^{-3} = \frac{1}{8} \), it can be rewritten as: \( \log_2 \frac{1}{8} = -3 \).

Key Concepts

Exponential Form

Exponential form is a way of expressing numbers and mathematical relationships using a base and an exponent. When we say something is in exponential form, it looks like this: \( a^b = c \). Here, \( a \) is the base, \( b \) is the exponent, and \( c \) is the result of raising the base to the power of the exponent.
For example:

• \( 8^{-1} = \frac{1}{8} \) shows that 8 raised to the power of -1 equals \( \frac{1}{8} \).
• \( 2^{-3} = \frac{1}{8} \) demonstrates that 2 raised to the power of -3 equals \( \frac{1}{8} \).

It's important to remember that a negative exponent, like in these examples, indicates a reciprocal. This makes exponential forms quite powerful in representing numbers concisely. Through understanding exponential form, we set the foundation to transform it into other forms, like logarithms.

Logarithms

Logarithms are mathematical operations that are the inverse of exponentiation. They help us express numbers by identifying the power needed to obtain a certain number using a given base. If you understand that \( a^b = c \), then \( \log_a c = b \).
In our examples:

• \( \log_8 \frac{1}{8} = -1 \), because raising 8 to the power of -1 gives \( \frac{1}{8} \).
• \( \log_2 \frac{1}{8} = -3 \), because 2 to the power of -3 equals \( \frac{1}{8} \).

Logarithms are incredibly useful in solving equations where the unknown is an exponent. By knowing how to switch between exponential and logarithmic forms, you can solve a wide range of problems with ease. This is why understanding both forms is crucial for progressing in mathematics.

Mathematical Transformations

Mathematical transformations involve changing an expression from one form to another while maintaining its value. In the context of logarithms and exponential forms, transformations allow us to switch between these two representations.
For example, transforming \( 8^{-1} = \frac{1}{8} \) into logarithmic form involves interpreting it as \( \log_8 \frac{1}{8} = -1 \). This transformation keeps the equality true, but changes how we interpret the relationship between the numbers.
To perform a transformation:

• Identify the base (\( a \)) and the result (\( c \)) in the exponential form.
• Rewrite using the logarithm where the base is \( a \) and the result is the logarithm equals the exponent \( b \).

Understanding these transformations solidifies your grasp of both exponential forms and logarithms. Using transformations, you can tackle various math problems more efficiently, especially those in algebra and higher-level mathematics.