Problem 13 - A Graphical Approach to College Algebra

Question

For each statement, write an equivalent statement in exponential form. $$\log _{\sqrt{3}} 81=8$$

Step-by-Step Solution

Answer

The equivalent exponential statement is \( (\sqrt{3})^8 = 81 \).

Step 1: Understand the Logarithmic Statement

The statement is \( \log_{\sqrt{3}} 81 = 8 \). It reads "the logarithm of 81 with base \( \sqrt{3} \) is 8". This means, if expressed in another form, the base \( \sqrt{3} \) raised to the power of 8 equals 81.

Step 2: Convert to Exponential Form

To convert the logarithmic statement \( \log_{\sqrt{3}} 81 = 8 \) into exponential form, rewrite it as \( (\text{base})^{(\text{result})} = \text{number} \). Thus, we have \( (\sqrt{3})^8 = 81 \).

Step 3: Verify the Conversion

Check that the exponential form correctly represents the original logarithmic statement by confirming \( (\sqrt{3})^8 = 81 \). Indeed, this conversion is valid because if you calculate \( (\sqrt{3})^8 \), it equals 81.

Key Concepts

Logarithms

Logarithms are a mathematical concept that helps us understand how numbers can be expressed as powers of each other. For any positive number greater than one, the logarithm measures how many times one number, called the base, should be multiplied by itself to obtain another number. For example, in the expression \( \log_{\sqrt{3}} 81 = 8 \), the logarithm tells us how many times \( \sqrt{3} \) must be multiplied by itself to get 81.
Logarithms are particularly useful in situations where numbers grow exponentially, such as in population studies or financial investments. They simplify complex multiplicative relationships by transforming them into easier-to-handle additions and subtractions.

• The base of the logarithm is an essential part; it is the number repeatedly multiplied.
• The logarithmic result indicates the power applied to the base to produce a given number.

Understanding this concept allows for the translation of logarithmic expressions to and from exponential ones, greatly aiding in various mathematical applications.

Exponential Form

Exponential form is another way to express numbers using powers. In the example, the statement \( \log_{\sqrt{3}} 81 = 8 \) can be converted into exponential form as \( (\sqrt{3})^8 = 81 \). This means that by multiplying \( \sqrt{3} \) eight times, you reach 81.
Expressions in exponential form like \( a^b \) indicate a base \( a \) raised to a power \( b \). This way of expressing numbers allows for easier computations and understanding of large values or growth patterns.

• "Base" refers to the number that is multiplied.
• "Exponent" or "power" shows how many times the base is used as a factor.

Converting between logarithmic and exponential forms is a key skill, allowing for the simplification and resolution of equations in algebra, calculus, and related disciplines.

Mathematical Reasoning

Mathematical reasoning involves logical thinking to analyze situations and solve problems. When dealing with logarithms and exponential forms, reasoning enables us to understand relationships between numbers and various mathematical entities.
In the conversion of \( \log_{\sqrt{3}} 81 = 8 \) to \( (\sqrt{3})^8 = 81 \), mathematical reasoning helps verify the correctness of this equivalence by re-examining the calculations done in exponential terms.

• Reasoning ensures that each step logically follows from the previous one.
• By understanding the concepts fully, one can explore further implications and applications.

Good mathematical reasoning applies not just to verifying calculations but also to predicting outcomes and forming strategies for more complex problems. It forms the bedrock of most mathematical problem-solving processes.