Problem 29 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \log _{6} 216 $$

Step-by-Step Solution

Answer

The value of \( \log_{6} 216 \) is 3.

Step 1: Understand the problem

The expression \( \log_{6} 216 \) is asking us to find the power to which the base \( 6 \) must be raised to yield 216. This means we are trying to solve for \( x \) in the equation \( 6^x = 216 \).

Step 2: Estimate and Test

First, make an educated guess by estimating values of \( x \) to see what power of 6 gets close to 216. We know that \( 6^2 = 36 \) and \( 6^3 = 216 \). Hence, \( 6^3 \) equals 216.

Step 3: Verify Solution

Check by calculating \( 6^3 \) explicitly to confirm our solution:\[ 6 \times 6 = 36 \] and \[ 36 \times 6 = 216 \].This verifies that \( 6^3 = 216 \), confirming our calculation.

Key Concepts

Understanding Logarithms

Logarithms might seem challenging at first, but they are extremely helpful for solving equations involving powers of numbers. A logarithm is essentially the inverse of an exponent. When you see an expression like \( \log_{b} a \), it asks: "To what power must the base \( b \) be raised, to result in \( a \)?" This question boils down to the equation \( b^x = a \). In our example, \( \log_{6} 216 \) asks, "What power of 6 gives us 216?" Logarithms help us to simplify complex equations by turning multiplication into addition, which is easier to manage mathematically.

• Logarithms answer "What power do we need?"
• They switch multiplication into addition when dealing with equations.
• Useful in many areas of math, including solving exponential growth problems.

Exponents and Their Role

Exponents are one of the fundamental concepts in mathematics and refer to the process of raising a number to a certain power. In the expression \( b^n \), \( b \) is the base and \( n \) is the exponent, meaning \( b \) is multiplied by itself \( n \) times. Exponents are not only a shortcut for repetitive multiplication but also provide a way to express large numbers efficiently. They are central to understanding logarithms, as they represent the problem logarithms aim to solve.

• An exponent indicates how many times a number (base) is multiplied by itself.
• Examples: \( 2^3 = 2 \times 2 \times 2 = 8 \) and \( 5^4 = 5 \times 5 \times 5 \times 5 = 625 \).
• Exponents simplify mathematical operations that involve large numbers.

Powers of Numbers

Powers of numbers are a way to express repeated multiplication of the same number. This concept is crucial in many areas of math, including solving equations and scientific notation. When we raise a number to a power, it results in an exponential growth, where small increases in the exponent lead to large increases in the result. Understanding powers is essential for grasping exponential functions, logarithms, and complex equations. In our specific example, \( 6^3 = 216 \) demonstrates how a relatively small base raised to a power results in a larger product.

• Powers denote repeated multiplication, like \( b^n \) or "\( b \) to the power of \( n \)".
• The equation \( 6^3 = 216 \) shows a base 6 raised to a power of 3.
• Powers model exponential growth and are vital in scientific and mathematical calculations.