Question

CRITICAL THINKING Evaluate each logarithm. (Hint: For each logarithm \(\log _b x\), rewrite \(b\) and \(x\) as powers of the same base.) a. \(\log _{125} 25\) b. \(\log _8 32\) c. \(\log _{27} 81\) d. \(\log _4 128\)

Short answer

The evaluated values of the logarithms are a. \(2/3\), b. \(5/3\), c. \(4/3\), d. \(7/2\).

Step-by-step solution

Evaluating logarithm a

In this logarithm, we have \(\log_{125} 25\). We can rewrite 125 and 25 as powers of 5, i.e., \(125 = 5^3\) and \(25 = 5^2\). Therefore, this logarithm becomes \(\log_{5^3} {5^2}\). Using the rule of logarithms \(\log_b {b^k} = k\), the answer is \(2/3\).

Evaluating logarithm b

In this logarithm, we have \(\log_8 {32}\). We can rewrite 8 and 32 as powers of 2, i.e., \(8 = 2^3\) and \(32 = 2^5\). Therefore, this logarithm becomes \(\log_{2^3} {2^5}\). Using the rule of logarithms \(\log_b {b^k} = k\), the answer is \(5/3\).

Evaluating logarithm c

In this logarithm, we have \(\log_{27} {81}\). We can rewrite 27 and 81 as powers of 3, i.e., \(27 = 3^3\) and \(81 = 3^4\). Therefore, this logarithm becomes \(\log_{3^3} {3^4}\). Using the rule of logarithms \(\log_b {b^k} = k\), the answer is \(4/3\).

Evaluating logarithm d

In this logarithm, we have \(\log_4 {128}\). We can rewrite 4 and 128 as powers of 2, i.e., \(4 = 2^2\) and \(128 = 2^7\). Therefore, this logarithm becomes \(\log_{2^2} {2^7}\). Using the rule of logarithms \(\log_b {b^k} = k\), the answer is \(7/2\).

Key concepts

Exponents

Exponents are a way to express numbers through repeated multiplication. Understanding exponents is crucial because they allow us to describe very large or very small numbers compactly. For example, the number 125 can be expressed as \(5^3\), which means that you multiply 5 by itself three times: \(5 \times 5 \times 5\). Similarly, the number 32 is expressed as \(2^5\).
By rewriting numbers in their exponential forms, we can make complex log problems much simpler. Spotting patterns in these exponent forms helps you tackle problems such as finding the base in a logarithmic equation more easily.
When working on logarithms, converting numbers into their base powers is your primary tool. Whether it's solving \(\log_8 32\) or \(\log_{27} 81\), this skill will simplify many steps.

Logarithmic Identities

Logarithmic identities are rules that govern the operations of logarithms. One important identity is \(\log_b {b^k} = k\). This means that if you have a logarithm with its base and power matching, the answer is simply the exponent. It simplifies calculations because you can skip directly to the result if the equation is set up in this log form.
When solving examples like \(\log_{5^3} {5^2}\), you directly utilize this identity. The answer for this is \(2/3\) because you apply \(k\) as \(2\) over the \(3\).
These identities serve as shortcuts that reduce complex logarithm problems into straightforward arithmetic, saving time and reducing errors. They keep the math elegant and concise.

Critical Thinking

Critical thinking in the context of math means approaching problems not just by following rules, but by truly understanding the concepts involved. Consider solving \(\log_{125} 25\).
Firstly, realize both 125 and 25 can be expressed with a common base, 5. This realization doesn't just help solve the problem, but enhances comprehension. It stretches thinking beyond fixed steps.
Critical thinking involves questioning: what if you used another base? Can all numbers be expressed similarly? This kind of thinking deepens your mathematical intuition and prepares you for varied problem types.

Mathematical Equivalence

Mathematical equivalence means that two expressions represent the same value, even if they look different. In logarithms, you're often looking to convert a tricky expression into one that's equivalent but easier to handle by matching it to known forms.
For instance, rewriting \(\log_{8} 32\) into \(\log_{2^3} 2^5\) makes the problem much more manageable. Despite the transformation, the math remains equivalent and accurate.

• This involves identifying equivalent forms of expressions.
• It tests flexibility in manipulating numbers while preserving their value.
• Recognizing equivalence allows you to apply known rules easily.

This depth of understanding is key. It makes problem-solving methodical and efficient, allowing you to find solutions confidently.