Problem 25 - Algebra for College Students

Question

Evaluate each logarithmic expression. \(\log _{6} 216\)

Step-by-Step Solution

Answer

\(\log_{6} 216 = 3\)

Step 1: Understand the Problem

We need to evaluate the logarithm \(\log_{6} 216\). This means we are looking for the power \(x\) such that \(6^x = 216\).

Step 2: Express 216 as a Power of 6

Try to express 216 as a power of 6 by factoring 216. Start dividing by 6 until the quotient is 1.\\[216 \div 6 = 36\] \\[36 \div 6 = 6\] \\[6 \div 6 = 1\] \Hence, \(216 = 6^3\).

Step 3: Calculate the Logarithm

Since we have expressed 216 as \(6^3\), it follows that \(\log_{6} 216 = \log_{6} 6^3\). By the power rule for logarithms, \(\log_{b}(b^a) = a\), we have:\\[\log_{6} 6^3 = 3\]

Step 4: Confirm the Solution

Double-check the calculations and ensure the factors of 216 yield the expression \(6^3\). Since everything checks out, the value of \(\log_{6} 216\) is confirmed as correct.

Key Concepts

Evaluate Logarithms

Evaluating a logarithm means finding the power to which a given base number must be raised to yield a certain value. For example, when evaluating \(\log_{6} 216\), we are determining which exponent, placed on 6, would result in the number 216. In simpler terms, if you imagine 6 as a superhero base number, you are seeking the power or the number of times it should multiply itself to transform into 216. By comparing the logarithmic expression with the power expression\[6^x = 216,\]an equation like this helps us understand that finding a logarithm just means figuring out this unknown exponent \(x\). Once you understand the relationship between logarithms and exponents, the process of evaluating becomes a lot more manageable.

Power Rule for Logarithms

The power rule for logarithms is a very handy mathematical principle. It allows you to simplify complex logarithmic expressions into simpler calculations. In general, the rule states:\[\log_{b}(b^a) = a.\]This means if you have a logarithm whose base matches the base of the exponent in the expression, its value is simply the exponent itself. Let's connect this to the original problem: when we expressed 216 as \(6^3\), it transformed our problem into:\[\log_{6} 6^3.\]Applying the power rule here, we see that the logarithm simply equals 3. This straightforward rule helps to remove the complexity of larger numbers and makes solving logarithmic problems like a friendly puzzle, where the answer often lies in knowing the exponent.

Factoring Numbers

Factoring numbers is about breaking down a number into its fundamental multiplicative components. In other words, take a number and express it as a product of smaller numbers, which when multiplied together, give the original number back. When solving logarithmic problems, factoring can help simplify expressions, just like it did for \(216\) into \(6^3\).

• Start with the original number, in this case, 216.
• Divide by the smallest prime number possible until you cannot divide anymore (for 216, divide by 6 repeatedly).
• These divisors (in this case, 6) will be your factors.

To clearly solve \(216 = 6^3\), start dividing:\[216 \div 6 = 36,\]\[36 \div 6 = 6,\]\[6 \div 6 = 1,\]showing \(216 = 6^3\). Factoring makes it easier to spot relationships between numbers and apply rules like the power rule in logarithms.