Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Express each equation in logarithmic form. $$2^{6}=64$$

Step-by-Step Solution

Answer

The logarithmic form of the given exponential equation \(2^{6}=64\) is: \(\log_{2}(64) = 6\).

Step 1: Identify the base, exponent, and result

In the exponential equation \(2^{6}=64\), we have a base of \(2\), exponent of \(6\), and result as \(64\).

Step 2: Express the equation in logarithmic form

Using the property \(\log_{a}(c)=b\), substitute the base, exponent, and result we found in Step 1. We get: \[\log_{2}(64) = 6\] And we are done. The logarithmic form of the given exponential equation is: \[\log_{2}(64) = 6\]

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear as exponents. For example, in the equation \( 2^{6} = 64 \), 2 is the base and 6 is the exponent. This tells us that we are multiplying 2 by itself 6 times. Exponential equations often describe how quantities increase rapidly over time, such as populations, investments, or radioactive decay.
Understanding exponential equations is essential because they appear frequently in various areas of science and engineering. An exponential equation like \( 2^{6} = 64 \) implies that the sequence of multiplications, or powers, results in 64.
Key characteristics of exponential equations include:

• They have a constant base raised to a variable exponent.
• Their solutions are often found through logarithmic transformations.
• They are non-linear, meaning the relationship is not a straight line when graphed.

Recognizing these can simplify conversion into other forms, like logarithmic form, which is commonly needed in problem-solving.

Logarithmic Form

The logarithmic form is another way to express exponential equations, focusing instead on the concept of powers or exponents. In the equation \( 2^{6} = 64 \), converting it to a logarithmic form involves identifying the base, exponent, and the result.
In logarithmic form, the equation is written as \( \log_{2}(64) = 6 \). This is because logarithms express the power or exponent needed for a base to become a specific number.
Logarithmic form serves as a powerful tool for solving equations where the unknown appears as an exponent:

• It reverses the exponentiation process by asking "to what power must the base be raised to produce the result?"
• Facilitates the simplification of equations, making it easier to solve for unknown variables.
• Used extensively in scientific calculations involving exponential growth or decay.

By representing exponential equations in this alternative form, we bridge intuition and computation, providing a deeper understanding of the multiplex relationship between variables.

Conversion between Exponential and Logarithmic Forms

The conversion between exponential and logarithmic forms hinges on understanding the relationship between exponents and logarithms. This conversion is fundamental in algebra and helps solve complex real-world problems.
For an exponential equation like \( a^{b} = c \), you convert it into a logarithmic form as \( \log_{a}(c) = b \). This tells us the exponent \( b \) needed for the base \( a \) to equal \( c \).
Key steps in conversion include:

• Identify the base, exponent, and resultant value from the exponential form.
• Apply the logarithmic transformation \( \log_{a}(c) = b \).
• Verify by solving the logarithmic equation to ensure consistency with the original calculation.

Such conversions are crucial because they transform complex multiplication problems into simpler additive terms easy to work with. As such, this is widely used in scenarios like calculating interest rates, understanding sound intensities, or in seismic activities, enhancing our capability to analyze data logarithmically.