Problem 19 - 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 \(\log _{2} 8=3.\) $$6^{-2}=\frac{1}{36}$$

Step-by-Step Solution

Answer

The logarithmic form of given equation is \(\log_{6}\frac{1}{36}=-2\).

Step 1: Identify Base, Exponent and Result

In the given equation \(6^{-2} = \frac{1}{36}\), '6' is the base, '-2' is the exponent and the result is \(\frac{1}{36}\).

Step 2: Convert to Logarithmic Form

Using the equivalence between exponential and logarithmic forms, this will turn into \(\log_{6}\frac{1}{36}=-2\).

Key Concepts

Understanding Exponential Equations

Exponential equations are mathematical expressions where a number, called the base, is raised to a power. This power is known as the exponent. In the expression \(6^{-2} = \frac{1}{36}\), the base is \(6\), and the exponent is \(-2\). Here, the equation shows that \(6\) raised to the power of \(-2\) results in \(\frac{1}{36}\), a fractional outcome. Exponential equations are used to represent growth or decay patterns, making them useful in diverse fields like finance, science, and engineering.

Exploring Base and Exponent

The base and the exponent are key components of exponential equations. The base is the number being multiplied by itself, and the exponent tells us how many times to do this multiplication. For instance:

• In \(6^{-2} = \frac{1}{36}\), \(6\) is the base, signifying the number you'll manipulate.
• The exponent \(-2\) indicates that you first find the reciprocal of the base, \(\frac{1}{6}\), due to the negative sign, and then square it.

Building comfort with identifying and interpreting the base and exponent is crucial for solving operational problems and converting between different forms of mathematical expressions.

Converting Equations

Converting equations helps us translate expressions into different forms, making them easier to work with or understand. In this concept, we focus on changing an exponential equation into its logarithmic form. This conversion allows us to express the relationship between the base, exponent, and result differently.

The purpose is to shift from multiplication to understanding powers in terms of logarithms. By doing so, you give insights into how elements of the equation relate. For example, if you have \(6^{-2} = \frac{1}{36}\), recognizing how these elements convert within the logarithmic context clarifies their mathematical meaning.

Mastering Logarithmic Conversion

Logarithmic conversion involves changing an exponential equation into a logarithmic form. This new form helps in understanding and solving equations that involve exponents. It's especially useful when dealing with complex mathematical problems.

• The general idea is to reframe \(b^e = R\) into \(\log_b R = e\).
• In our example, \(6^{-2} = \frac{1}{36}\) converts to \(\log_{6} \frac{1}{36} = -2\).

This conversion is helpful because logarithms are involved in studying growth rates and solve exponential equations systematically. Grasping logarithmic conversion broadens your mathematical toolkit, allowing you to solve equations with greater ease.