Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in logarithmic form. \(10^{a}=b\)

Step-by-Step Solution

Answer

\(\log_{10} b = a\)

Step 1: Understand the Definition

Logarithmic form is the inverse of exponential form. If you have an equation in exponential form like \(a^x = b\), the equivalent logarithmic form is \(\log_a b = x\).

Step 2: Identify Components of the Exponential Equation

In the given exponential equation \(10^a = b\), identify the base, exponent, and the result: the base is 10, the exponent is \(a\), and the result is \(b\).

Step 3: Reapply Definitions to Rewrite the Logarithmic Form

Using the identified components and the definition of logarithms, rewrite the equation \(10^a = b\) in logarithmic form. The base 10 translates to the base of the logarithm, \(a\) becomes the result of the logarithmic operation, and \(b\) becomes the argument. Hence, the logarithmic form is \(\log_{10} b = a\).

Key Concepts

Exponential Equation

An exponential equation is a type of mathematical equation that involves an expression where a number, known as the base, is raised to a power, which is represented by the exponent. This is usually written in the form \( a^x = b \), where \( a \) (the base) is a constant, \( x \) (the exponent) is the unknown variable, and \( b \) is the result of the power operation. Exponential equations are important because they are used to describe phenomena that show exponential growth or decay, such as population growth, radioactive decay, and compound interest. To solve an exponential equation, you often need to manipulate it into a form where the exponents can be compared or converted, such as switching to a logarithmic form. This conversion relies on the understanding that logarithms form the inverse operations of exponentials.

Inverse Operations

Inverse operations are pairs of mathematical operations that undo each other. In the context of exponential and logarithmic equations, the concept of inverse operations is central. Exponentiation and logarithms are inverse operations. This means that performing one operation can "undo" the effect of the other. For example:

• If you have \( 2^3 = 8 \), then to find the power or to "undo" the exponentiation, you use the logarithm: \( \log_2 8 = 3 \).

Understanding inverse operations is crucial when dealing with exponential or logarithmic equations because it allows us to solve these types of equations and convert one form into another. By recognizing that the operations are reverses of one another, we can effectively isolate variables and find solutions.

Logarithmic Equation

A logarithmic equation is an equation that involves the logarithm of a certain number. It is expressed in the form \( \log_a b = x \), where \( a \) is the base of the logarithm, \( b \) is the number you're taking the log of (known as the "argument"), and \( x \) represents the power to which the base must be raised to obtain \( b \).Logarithmic equations are particularly useful when dealing with very large or small numbers, as they can simplify the arithmetic operations involved. To solve a logarithmic equation, you often need to use properties of logarithms, such as transforming the equation back into its exponential form to make it easier to solve. This involves applying the inverse operations principle, recognizing that if \( a^x = b \), then \( \log_a b = x \) by definition.

Base and Exponent Identification

In both exponential and logarithmic forms, identifying the base and the exponent is a critical first step. In exponential form, such as \( 10^a = b \), the base is the number 10 and the exponent is \( a \). It’s the power to which the base is raised to produce the number \( b \).When translating to logarithmic form, identifying these components helps in correctly setting up the new equation. Here’s how it translates:

• The base from the exponential form becomes the base in the logarithm: \( \log_{10} \).
• The result of the exponential operation, \( b \), becomes the argument of the logarithm.
• The exponent \( a \) in the exponential form becomes the result of the logarithmic equation: \( \log_{10} b = a \).

Therefore, clear identification of these components ensures accurate conversion between the forms, which is key to solving and understanding these equations.