Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Give the exact answer. $$\log _{3} 81=x$$

Step-by-Step Solution

Answer

x = 4

Step 1: Understand the log equation

The equation given is \( \log_3 81 = x \). This means we are looking for a number \( x \) such that when 3 is raised to the power of \( x \), the result is 81. In mathematical form, this is equivalent to \( 3^x = 81 \).

Step 2: Express 81 as a power of 3

To solve \( 3^x = 81 \), we should express 81 as a power of 3. Notice that 81 can be written as \( 3^4 \) because \( 3 \times 3 \times 3 \times 3 = 81 \).

Step 3: Set the exponents equal

Since we have \( 3^x = 3^4 \), according to the equality of exponential functions with the same base, the exponents must be equal. Thus, \( x = 4 \).

Step 4: Conclusion

We've determined through these steps that the value of \( x \) that satisfies the original equation \( \log_3 81 = x \) is 4.

Key Concepts

Exponential Equations

Exponential equations are equations where variables appear as exponents. These equations are fundamental in mathematics as they model scenarios involving exponential growth or decay. Understanding exponential equations means grasping how the base and the exponent work together to create values. The general form of an exponential equation is \( a^x = b \), where \( a \) is the base and \( x \) is the exponent.
To solve exponential equations, find the same base on both sides of the equation. If both sides of the equation can be expressed with the same base, you can directly equate the exponents. For example, in the equation \( 3^x = 81 \), by recognizing 81 as \( 3^4 \), we can deduce that \( x = 4 \).

• Identify whether the equation can be expressed with the same base on each side.
• Use properties of exponents to simplify the expressions if possible.
• If bases are identical, equate the exponents to solve for the unknown variable.

Mastering these steps in solving exponential equations will make solving logarithmic equations straightforward, as you often encounter solving for unknown exponents in a similar manner.

Powers and Exponents

Powers and exponents express repeated multiplication of the same number. An expression like \( a^n \) tells us that \( a \) is multiplied by itself \( n \) times. Here, \( a \) is called the base, and \( n \) is called the exponent or power.
Powers help simplify complex equations by reducing long forms of multiplication to a more compact notation. For example, instead of writing \( 3 \times 3 \times 3 \times 3 \), we can simply write \( 3^4 \).

• When multiplying powers with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
• When dividing powers with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• A power raised to another power multiplies the exponents: \( (a^m)^n = a^{m \times n} \).

These principles help in simplifying exponential equations and transitioning into logarithmic concepts, allowing us to backtrack from powers (like \( 81 = 3^4 \)) to their roots.

Base of a Logarithm

A logarithm answers the question: "To which exponent must we raise the base to obtain a certain number?". In the expression \( \log_b a = x \), \( b \) is the base of the logarithm. It signifies the number that needs to be raised to a power (\( x \)) to yield a particular outcome (\( a \)).
The base is the critical part of understanding logarithms as it contextualizes the measure by which a certain number is evaluated. For example, in \( \log_3 81 = x \), the base is 3, indicating 81 is being evaluated in terms of powers or multiples of 3.

• A log with base 10 is known as a common logarithm, denoted simply as \( \log \).
• A log with base \( e \) (approximately 2.718) is a natural logarithm, denoted as \( \ln \).
• Changing the base of a logarithm can be done using the change of base formula: \( \log_b a = \frac{\log_c a}{\log_c b} \).

Comprehending the base within logarithms allows for seamless transition from logarithmic expressions to their exponential form, making problems like \( \log_3 81 = 4 \) intuitive once the connection to exponents is appreciated.