Problem 14 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{1.2} 1.44=2 $$

Step-by-Step Solution

Answer

\( 1.2^2 = 1.44 \)

Step 1: Understanding the Logarithmic Equation

The equation given is \( \log_{1.2} 1.44 = 2 \). This can be interpreted as saying: 1.2 raised to which power results in 1.44. The answer is 2, meaning \( 1.2^2 = 1.44 \).

Step 2: Converting Logarithmic Form to Exponential Form

The logarithmic equation \( \log_{1.2} 1.44 = 2 \) can be rewritten in exponential form as follows: The base 1.2, when raised to the power of 2, equals 1.44. Thus, the equation is \( 1.2^2 = 1.44 \).

Step 3: Write the Final Exponential Equation

After converting the logarithmic form, the exponential equation is \( 1.2^2 = 1.44 \).

Key Concepts

Logarithmic Equations

Logarithmic equations are a fundamental concept in mathematics. They express the relationship between a base, a number, and an exponent. Understanding these equations is crucial because they help solve problems that involve exponential growth or decay. A logarithmic equation involves a logarithm, which is essentially the inverse of exponentiation.

In simpler terms:

• The logarithm of a number is the exponent to which the base must be raised to obtain that number.
• The base is a fixed number greater than zero, and it cannot be one.
• The result or quotient represents the exponent itself.

For instance, in the equation \( \log_{1.2} 1.44 = 2 \), the base is 1.2. The logarithm defines that if you raise 1.2 to the power of 2, you will obtain 1.44. This foundational understanding makes it easier to manipulate these equations for further mathematical operations, such as conversions or solving for unknowns.

Conversion to Exponential Form

Converting a logarithmic equation to an exponential form is an essential process, making equations easier to understand and work with.

When we convert a logarithmic equation to its exponential form, we directly apply the definition of a logarithm:

• Given \( \log_b a = c \), the equivalent expression in exponential form is \( b^c = a \).
• This means that the base \( b \), raised to the exponent \( c \), equals the number \( a \).

The key to conversion is recognizing the components of the logarithm: the base, the number, and the logarithm's result, which is the exponent in exponential form.

For the example \( \log_{1.2} 1.44 = 2 \), converting to exponential form involves writing it as \( 1.2^2 = 1.44 \), clearly showing how the logarithmic equation claims that 1.2 raised to the power of 2 equals 1.44.

This process notably simplifies the equation and is useful for validating solutions or solving for unknown components in complex problems.

Base and Power Relationship

In mathematics, particularly in exponents and logarithms, understanding the relationship between the base and the power is imperative. This foundation helps when dealing with equations that involve growth, decay, or change over time.

To grasp this relationship:

• The base is the number that is being multiplied by itself. It is the core of exponential expressions.
• The power, or exponent, demonstrates how many times the base is used in the multiplication.
• The result from the base raised to the power describes a quantity or value obtained.

For example, in the equation \( 1.2^2 = 1.44 \), the base is 1.2, and the exponent is 2. This means that 1.2 is multiplied by itself once, resulting in 1.44.

Understanding this connection helps students appreciate the simplicity yet the power of exponential and logarithmic expressions. It's particularly useful when trying to decipher equations or when needing to form equations in various mathematical applications.