Problem 18 - College Algebra

Question

Write each equation in its equivalent logarithmic form. $$ b^{3}=343 $$

Step-by-Step Solution

Answer

\(\log 343 = 3\)

Step 1: Identify the Base, Exponent, and Result

The given equation is in the form of \(a^b = c\). Here, \(a = b\), \(b = 3\), and \(c = 343\)

Step 2: Apply the Logarithmic Form Conversion

Now, convert the equation from exponential form \(a^b = c\), to logarithmic form \(\log_a c = b\) to obtain \(\log_b 343 = 3\)

Step 3: Simplify the Logarithmic Equation

Since the base is not specified in the logarithmic form, it's understood to be base 10. Rewrite the equation as \(\log_{10} 343 = 3\). However, since it's commonplace to simply write such logarithms without the base when the base is 10, the final answer is simply \(\log 343 = 3\).

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve logarithms. They often require you to find unknown variables embedded in a logarithm. To understand logarithmic equations, an essential realization is that they are the inverses of exponential equations.
For instance, if you have an exponential equation like \(b^3 = 343\), the related logarithmic form would be \(\log_b 343 = 3\). This transformation reflects the relationship that the logarithm provides the exponent in an exponential expression. In other words, if amplifying a base to a specific power reaches a certain result, the logarithm gives the power originally used.

When solving logarithmic equations, remember to apply inverse operations carefully and ensure you verify your solutions by plugging them back into the original equation. This way, you confirm their correctness and avoid errors.

Exponentiation

Exponentiation involves raising a number, known as the base, to a specific power, called the exponent. It is expressed as \(a^b = c\), where \(a\) represents the base, \(b\) is the exponent, and \(c\) is the product of this operation.
In the equation \(b^3 = 343\), \(b\) is raised to the power of 3 to yield 343. This result means the base \(b\) is multiplied by itself three times to produce the number 343.

It's beneficial to thoroughly grasp this concept because many mathematical operations and real-world applications involve exponentiation, like calculating compound interest or understanding geometric growth patterns.

Base of a Logarithm

The base of a logarithm is pivotal to its definition and calculation. In a logarithmic expression denoted as \(\log_a c = b\), \(a\) is known as the base. It specifies the number that is raised to power \(b\) to yield \(c\).

In the exponential expression \(b^3 = 343\), switching to a logarithmic expression gives us \(\log_b 343 = 3\). Here, \(b\), the value being exponentially manipulated, becomes the base of the logarithm.
A crucial note: often, when no base is mentioned in a logarithm, it is assumed to be 10, which is known as the common logarithm. The common base makes computations and applications in many standard settings simpler and more convenient.