Problem 92 - Algebra 2

Question

What is the logarithmic form of the exponential equation \(2^{4}=16 ?\) $$ \mathrm{F} . \log _{16} 2=4 \quad \text { G. } \log _{16} 4=2 \quad \text { H. } \log _{4} 16=2 \quad \text { J. } \log _{2} 16=4 $$

Step-by-Step Solution

Answer

The correct answer is J. \(\log_{2}(16) = 4\).

Step 1: Understand the conversion rule

The rule for converting an exponential equation to logarithmic form is \(b^{p}=q\) can be written as \(\log_{b}(q)=p\), where b is the base, p is the power or exponent and q is the result.

Step 2: Apply the conversion rule

Applying this rule to the given exponential equation \(2^{4}=16\), the base b=2, the power p=4, and the result q=16. Therefore, the logarithmic form will be \(\log_{2}(16) = 4\).

Key Concepts

exponential equations

Exponential equations are mathematical expressions where numbers are raised to a power. For example, in the equation \(2^4=16\), 2 is called the base, and 4 is the exponent. Exponents indicate how many times a number, the base, is multiplied by itself. In this case, 2 is multiplied by itself four times, resulting in 16. Exponential equations often appear in different scientific contexts because they can model exponential growth or decay, like population increase or radioactive decay. Recognizing the parts of an exponential equation is crucial for converting it into other forms, like the logarithmic form.

logarithms

Logarithms are the inverse operations of exponents. If an exponential equation is known, the logarithm can be used to find the exponent when the base and the result are known. - For example, in the exponential equation \(2^4=16\), the logarithm helps to find the relation between the base 2 and the result 16, revealing the exponent 4. - The concept behind logarithms is asking the question, "To what power must a given base be raised, to produce a certain number?" In the logarithmic form, this is represented as \( \log_2(16) = 4 \), which directly translates the exponential equation into a form that can be more convenient for solving certain types of problems. Logarithms are extensively used in various fields such as engineering, biology, and finance due to their ability to simplify multiplication and division into addition and subtraction.

conversion rules

Conversion rules help change an exponential equation into a logarithmic form smoothly. - The key to conversion is understanding the basic relationship: \(b^p=q\) is translated into \( \log_b(q)=p \).- Here, \( b \) is the base, \( p \) is the power or exponent, and \( q \) is the result.To convert the exponential equation \( 2^4 = 16 \) into its logarithmic form, identify the base \( b \) as 2, the exponent \( p \) as 4, and the result \( q \) as 16. Applying the conversion rule, this becomes \( \log_2(16) = 4 \). Understanding and applying the conversion rules are important for students to move between different numerical representations effectively. The familiarity with conversion techniques helps in understanding complex mathematical concepts and solving real-life problems more comfortably.