Problem 11 - A Graphical Approach to Precalculus with Limits

Question

For each statement, write an equivalent statement in exponential form. Do not use a calculator. $$\log \sqrt{3} 81=8$$

Step-by-Step Solution

Answer

\( (\sqrt{3})^8 = 81 \)

Step 1: Identify the logarithmic form

The given statement is \( \log_{\sqrt{3}} 81 = 8 \). This is in the form \( \log_b a = c \), where \( b = \sqrt{3} \), \( a = 81 \), and \( c = 8 \).

Step 2: Write the exponential form

The equivalent exponential form of \( \log_b a = c \) is \( b^c = a \). Thus, we write \( (\sqrt{3})^8 = 81 \).

Step 3: Simplify the base

Express the square root as an exponent: \( \sqrt{3} = 3^{1/2} \). Substitute to get \((3^{1/2})^8 = 81 \).

Step 4: Simplify the expression

Use the power of a power rule \((a^m)^n = a^{mn}\) to simplify: \( 3^{1/2 \times 8} = 3^{4} \). So, \( 3^4 = 81 \). Confirm that both sides equal, which they do.

Key Concepts

Logarithmic Form

Logarithmic form is a way to express the relationship between a base, an exponent, and a resultant number. In the notation \( \log_b a = c \), \( b \) represents the base, \( a \) is the number you're taking the logarithm of, and \( c \) is the exponent or power to which the base must be raised to produce \( a \). This form is essential for solving exponential equations, as it helps to identify the exponent needed for a base to result in a specific number.

In our exercise, the logarithmic statement is \( \log_{\sqrt{3}} 81 = 8 \). From this, we can deduce that when the base \( \sqrt{3} \) is raised to the power of 8, it should equal 81. This process of converting a logarithmic expression into an exponential one can help solve various mathematical problems where direct calculation is not possible.

Exponents

Exponents are a fundamental concept in mathematics, representing how many times a number, known as the base, is multiplied by itself. In expressions like \( b^c \), \( b \) is the base and \( c \) is the exponent. Understanding how to work with exponents is crucial for dealing with exponential equations and logarithmic expressions.

When converting logarithmic form into exponential form, as we did with \( \log_{\sqrt{3}} 81 = 8 \) to \( (\sqrt{3})^8 = 81 \), we leverage our understanding of exponents. Here, \( \sqrt{3} \) is written as \( 3^{1/2} \), and the exponential form becomes \( (3^{1/2})^8 \), which simplifies to \( 3^4 \). This process demonstrates the power of the exponentiation rules, like the power of a power rule \((a^m)^n = a^{mn}\), which simplify calculations and show the relationship between numbers.

Exponential Equations

Exponential equations are equations in which variables appear as exponents. Solving these types of equations often involves expressing them in simpler forms, like what we did in the exercise with logarithmic and exponential forms. By converting the logarithmic expression \( \log_{\sqrt{3}} 81 = 8 \) into its equivalent exponential form \( (\sqrt{3})^8 = 81 \), we paved the way to solving for the unknown.

The simplification process for exponential equations often employs rules like the power of a power: \( (a^m)^n = a^{mn} \). In our exercise, this rule allowed us to simplify \( (3^{1/2})^8 = 3^{4} \), making it clear that \( 3^4 = 81 \). Such techniques are invaluable when tackling exponential equations, as they transform complex expressions into more manageable forms, which can be used to verify solutions or further solve problems.