Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3.\) $$8^{2}=64$$

Step-by-Step Solution

Answer

\(\log _{8} 64=2\)

Step 1: Identify the base, exponent and result

In the exponential equation \(8^{2}=64\), 8 is the base, 2 is the exponent and 64 is the result.

Step 2: Write down the logarithmic form

Using the base, exponent and result identified in step 1, put these into logarithmic form. The base becomes the base of the logarithm, the result becomes the argument and the exponent becomes the value of the logarithm.

Step 3: Substitute the values

Substitution gives \(\log _{8} 64=2\).

Key Concepts

Exponential Equations

Exponential equations are a fundamental concept in mathematics, involving expressions where variables appear as exponents. Understanding these equations is crucial as they frequently occur in various scientific and real-life problem-solving scenarios. An exponential equation is generally in the form of \(b^n = x\), where:

• \(b\) is the base,
• \(n\) is the exponent,
• \(x\) is the result (or power).

For the exercise \(8^2 = 64\), 8 is the base, 2 is the exponent, and 64 is the result. Recognizing these components helps in transforming the equation into its logarithmic form. Exponential equations can often be challenging, but with practice, identifying these parts becomes easier.

Logarithms

Logarithms are the inverse operation of exponentiation. They provide a way to solve equations where the variable is an exponent, making complex calculations more manageable. The core idea is that a logarithm answers the question: "To what power must a certain base be raised, in order to get a specific number?"

The general form of a logarithm is \(\log_b(x) = n\), meaning the power \(n\) that the base \(b\) must be raised to, in order to obtain \(x\). Applying this to our example, we see that the logarithmic form of \(8^2 = 64\) is \(\log_8(64) = 2\).

• The base \(8\) of the exponential equation becomes the base of the logarithm.
• The result \(64\) becomes the argument of the logarithm.
• The exponent \(2\) becomes the value of the logarithm.

Utilizing logarithms allows for easier computations, especially when dealing with large numbers or complex operations.

Mathematics

Mathematics is a diverse and extensive field that includes structures and concepts such as logarithms and exponential equations. These tools are essential for various applications including, but not limited to, financial modeling, scientific computations, and data analysis.

Mathematical understanding involves both theoretical knowledge and practical problem-solving skills. Concepts like exponential equations and logarithms are interconnected and highlight the beauty of mathematical symmetry.

• Recognizing patterns and relationships in mathematics can simplify seemingly complex problems.
• Practical exercises help solidify theoretical understanding, making abstract concepts more tangible.
• Mathematics teaches critical thinking and analytical skills.

As with the transformation of exponential equations into logarithmic form, consistent practice and exploration of mathematical ideas open up a world of discovery and innovation.