Problem 18 - 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 \(\log _{2} 8=3.\) $$9^{3 / 2}=27$$

Step-by-Step Solution

Answer

\(\log _{9} 27 = 3/2\)

Step 1: Identify the base, exponent and result in the exponential form

Looking at the given exponential equation \(9^{3 / 2} = 27\), notice that the base is '9', the exponent is '3/2' and the result is '27'

Step 2: Rewrite the exponential equation into logarithmic form

Using the identified base, exponent and result, rewrite the equation in the form of \(\log_{base} result = exponent\). So, place the base of 9 inside the log function, the result of 27 becomes what you are taking the log of, and set this equal to the exponent fraction 3/2. This converts the equation into logarithmic form, yielding \(\log _{9} 27 = 3/2\).

Key Concepts

Exponential Equation

An exponential equation is a mathematical statement where a quantity grows rapidly by multiplying a fixed number, termed the base, raised to the power of an exponent. These equations often appear in various mathematical, financial, and scientific contexts.

Key characteristics include:

• The base: the constant number that is raised to a power.
• The exponent: the power to which the base is raised, determining the extent of the multiplication process.
• The result: the outcome of the base raised to the exponent.

To illustrate, take the equation \(9^{3/2} = 27\). Here, '9' is the base, '3/2' is the exponent, and '27' is the result of this exponential expression.

Understanding exponential equations is crucial for solving many logarithmic problems, deriving exponential functions, and converting between exponential and logarithmic forms.

Base of a Logarithm

The base of a logarithm plays a vital role in the structure of a logarithmic equation. It indicates which number is repeatedly multiplied to reach the result that the logarithm refers to.

Consider the logarithmic form \(\log_{9} 27 = 3/2\). In this context, '9' serves as the base of the logarithm. It signifies that the number 9 is used as a factor, repeated in a way defined by the exponent '3/2', to produce the result '27'.

Some commonly used bases are:

• Base 10 (also called the common logarithm, written as \(\log\))
• Base \(e\) (the natural logarithm, written as \(\ln\)) where \(e\) is an irrational number approximately equal to 2.718.

Understanding the impact of the logarithm's base is essential for interpreting the scale and growth of the mathematical relationships they describe.

Exponent

The exponent in a mathematical expression indicates how many times the base is used as a factor. It's essentially a multiplier for the base, directing how it should expand.

In the example \(9^{3/2} = 27\), the exponent is '3/2'. This fractional exponent implies both a power and a root:

• The numerator '3' suggests that we cube the number 9.
• The denominator '2' suggests that we take the square root of 9 after cubing it.

Combining these operations leads to the final result. Exponents can be fractional, whole numbers, positive, or negative, each influencing how the base is manipulated<.br>
Understanding exponents is vital for converting between exponential and logarithmic forms, and solving complex equations.

Logarithmic Function

A logarithmic function is essentially the inverse of an exponential function. It answers the question: to what power must a base be raised to yield a specific number.

For example, consider the logarithmic equation \(\log_{9} 27 = 3/2\). This function tells us that the base '9', when raised to the power '3/2', results in '27'. In simpler terms:

• The logarithmic function \(\log_{b} x = y\) asks "What power \(y\) must base \(b\) be raised to, in order to get \(x\)?"

Logarithms are used in various fields, including science, engineering, and finance, to model exponential relations and solve equations where the variable is in the exponent.

By converting exponential equations into their logarithmic form, like in the given exercise, you can more easily solve for unknowns and comprehend their underlying properties.