Problem 257 - Calculus Volume 1

Question

Write the equation in equivalent logarithmic form. \(\quad 9^{0}=1\)

Step-by-Step Solution

Answer

The equivalent logarithmic form is \(\log_9{1} = 0\).

Step 1: Identify the Form

The given equation is in exponential form: \(a^b = c\), where \(a = 9\), \(b = 0\), and \(c = 1\).

Step 2: Understand the Logarithmic Form

The logarithmic form related to the equation \(a^b = c\) is \(\log_a{c} = b\).

Step 3: Substitute Values into Logarithmic Form

Substitute the values from the exponential form into the logarithmic form equation: \(\log_9{1} = 0\).

Step 4: Final Step: Review the Converted Form

The equation \(9^0 = 1\) in logarithmic form is \(\log_9{1} = 0\).

Key Concepts

Exponential Equations

Exponential equations involve expressions where a constant base is raised to a variable exponent. In the simplest form, they follow the pattern \( a^b = c \), where \( a \) is the base, \( b \) is the exponent, and \( c \) is the result of the power. Exponential equations can model growth or decay processes in real life, such as population growth or radioactive decay. Recognizing an exponential equation is crucial for converting it into its logarithmic form for easier solving or understanding of certain equations.Key points to remember about exponential equations:

• The base \( a \) is a constant number.
• The exponent \( b \) can be any real number and is what makes this an equation with potentially infinite results.
• The result \( c \) shows what we obtain when the base is multiplied by itself the number of times specified by the exponent.

Understanding exponential equations helps in identifying and converting them into other forms, especially logarithmic forms. These conversions are often essential in solving complex problems in mathematics.

Logarithmic Equations

Logarithmic equations are expressions where a logarithm, representing an exponent, is set equal to a number. They can be derived from exponential equations by solving for the exponent. In other words, if you have an equation like \( a^b = c \), you can convert it to a log equation like \( \log_a{c} = b \).Logarithms transform multiplicative relationships into additive ones, making complex calculations more manageable. Here's how they work in the context of the original exercise:- The base of the logarithm \( a \) is the same as the base of the exponent in the exponential form.- The result \( c \) from the exponential equation becomes the argument of the logarithm.- The exponent \( b \) from the exponential equation turns into the result of the logarithm.For example, converting \( 9^0 = 1 \) into logarithmic form results in \( \log_9{1} = 0 \). This equivalence shows that the exponent you raise 9 to in order to get 1 is 0.Comprehending logarithmic equations is fundamental to grasp transformations in equations and is also crucial for solving exponential problems in an alternative way.

Properties of Logarithms

Logarithms come with specific properties that can simplify algebraic expressions and solve equations efficiently. Understanding these properties enhances your ability to manipulate exponential and logarithmic forms interchangeably.Some important properties of logarithms include:

• Product Rule: \( \log_b{(XY)} = \log_b{X} + \log_b{Y} \) – The logarithm of a product is the sum of the logarithms.
• Quotient Rule: \( \log_b{\left(\frac{X}{Y}\right)} = \log_b{X} - \log_b{Y} \) – The logarithm of a quotient is the difference of the logarithms.
• Power Rule: \( \log_b{(X^n)} = n \cdot \log_b{X} \) – The logarithm of a power is the exponent times the logarithm of the base.
• Change of Base Formula: \( \log_b{X} = \frac{\log_c{X}}{\log_c{b}} \) – Allows conversion of logs from one base to another.

These properties not only make calculations easier but also enable more straightforward simplification of complex logarithmic expressions. Recognizing and utilizing these properties can streamline solving log equations, especially when dealing with large numbers or when the given base is inconvenient.