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Chapter 12: Problem 11

Simplify. $$ \log _{2} 16 $$

Short Answer

Expert verified
\(\text{log}_2 16 = 4\)

Step by step solution

01

Understand the Logarithm

A logarithm \(\text{log}_a b\) asks the question: 'To what power must \(a\) be raised, to yield \(b\)?' Here, \(a = 2\) and \(b = 16\).
02

Express the Number as a Power of the Base

Convert \(16\) to a power of \(\text{base}\) (which is \(2)\). \text{Since} \(2^4 = 16\), \(16\) can be written as \(2^4\).
03

Set Up the Logarithmic Expression

Using the relationship from Step 2, rewrite the logarithmic expression as \(\text{log}_2 (2^4)\).
04

Apply the Logarithm Power Rule

Utilize the logarithm power rule which states \(\text{log}_a (a^b) = b\). Therefore, \(\text{log}_2 (2^4) = 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Logarithms
Let鈥檚 start with simplifying logarithms. When you see a logarithmic expression, it might look a bit complicated, but it doesn鈥檛 have to be! The logarithm \( \text{log}_a b \) actually answers the question: 'To what power must our base \(a\) be raised to get \(b\)?'
In the example \( \text{log}_2 16 \), we need to determine what power 2 must be raised to in order to equal 16.
Let鈥檚 look at the number 16. To find an answer, we convert it into an exponential form or a power of 2.
Imagine it like this:
  • 2 raised to 4 equals 16: \(2^4 = 16\).
  • This makes finding the solution easier, as we can now rewrite the logarithmic expression in a simpler form: \( \text{log}_2 (2^4) \).

This becomes much simpler to handle in the subsequent steps!
Logarithmic Power Rule
Now, let's dig deeper into the logarithmic power rule, which is a key concept to simplify our expression further.
The power rule for logarithms states that \( \text{log}_a (a^b) = b \). In simple words, if you have a log of a number that is a base raised to a power, the result is the exponent itself!
Practical examples are always helpful: Consider our expression \( \text{log}_2 (2^4) \). According to the power rule, since the base \(a\) is 2 and the number we have is \(2^4\), we simply take the exponent:
  • This means \( \text{log}_2 (2^4) \) equals just 4.

Essentially, the logarithm function and the exponential function cancel each other out, leaving us with the power/exponent part.
Converting Numbers to Powers
For better comprehension, converting numbers to powers is crucial. Recognizing how to express a number as a power of another can make logarithmic problems a breeze.
Let鈥檚 consider our number 16 again. To express it as a power of 2, we need to find some power of 2 that equals 16:
  • We know that 2*2 equals 4 (or \(2^2 = 4 \)).
  • Similarly, 2 raised to the power of 3 equals 8 (or \(2^3 = 8\)).
  • Finally, multiplying one more time, 2 raised to the power of 4 equals 16 (or \(2^4 = 16\)).

This logical breakdown helps in easily converting numbers to a particular base power and simplifies the logarithmic expressions.
Understanding this thoroughly will make logarithm simplification second nature!

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Most popular questions from this chapter

Graph both equations using the same set of axes: $$ y=\left(\frac{3}{2}\right)^{x}, \quad y=\log _{3 / 2} x $$

How could you convince someone that $$\log _{a} c \neq \log _{c} a ?$$

Find the domain of each function. $$f(x)=\frac{x}{(x-2)(x+3)}$$

Solve by completing the square. $$ x^{2}+8 x=1 $$

Simplify. $$ \log _{1 / 4} \frac{1}{64} $$
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