Problem 17 - College Algebra

Question

For the following exercises, rewrite each equation in logarithmic form. $$ c^{d}=k $$

Step-by-Step Solution

Answer

The logarithmic form is \(d = \log_c{k}\).

Step 1: Understand the problem

We need to convert the equation \(c^d = k\) from its exponential form to its logarithmic form. The goal is to express the same relationship using a logarithm.

Step 2: Identify the components

Identify the 'base', the 'exponent', and the 'result' in the equation \(c^d = k\). Here, \(c\) is the base, \(d\) is the exponent, and \(k\) is the result.

Step 3: Apply the logarithmic rule

Recall the logarithmic form rule: If \(c^d = k\), then it can be rewritten in logarithmic form as \(d = \log_c{k}\). This rule states that the exponent \(d\) is equal to the logarithm of the result \(k\) to the base \(c\).

Step 4: Write the equation in logarithmic form

Now, using the identified components and the rule, rewrite \(c^d = k\) into logarithmic form: \(d = \log_c{k}\). This expresses \(d\) as the power to which the base \(c\) must be raised to obtain \(k\).

Key Concepts

Understanding Exponential Form

Exponential form is a way of expressing numbers using a base and an exponent. It is a concise method to denote repeated multiplication of a number by itself. For instance, in the equation \(c^d = k\), \(c\) is the base, and \(d\) is the exponent. This means that \(c\) is multiplied by itself \(d\) times to get \(k\). Exponents simplify complex multiplication expressions by reducing them to simpler terms.It's essential to understand the components of an exponential expression:

• Base: The number that is multiplied by itself.
• Exponent: Tells how many times the base is used in the multiplication.
• Result: The outcome of raising the base to the power of the exponent.

This form is prevalent in mathematics because it provides an efficient way to express large numbers and perform calculations involving repeated multiplication.

Decoding Logarithmic Rules

Logarithms are the inverse of exponentiation, helping to 'reverse' exponential calculations. An understanding of logarithmic rules is crucial when converting exponential equations into logarithmic form. When you have an equation of the form \(c^d = k\), its logarithmic form is \(d = \log_c{k}\). This illustrates that the exponent \(d\) is equal to the logarithm of \(k\) with base \(c\).Key logarithmic rules include:

• Logarithm of a Product: \(\log_b(xy) = \log_b{x} + \log_b{y}\)
• Logarithm of a Quotient: \(\log_b\left(\frac{x}{y}\right) = \log_b{x} - \log_b{y}\)
• Logarithm of a Power: \(\log_b(x^y) = y \cdot \log_b{x}\)

These rules help to break down complex logarithmic expressions into simpler ones, similar to how arithmetic rules simplify calculations.

Base and Exponent in Logarithms

In logarithms, understanding the roles of base and exponent is essential for solving equations. The base \(c\) in a logarithm, such as \(\log_c{k}\), correlates directly to the base in exponential form \(c^d = k\). It determines the progression of numbers that fit the logarithmic system.The exponent, translated as the logarithm's value, reveals how many times the base needs multiplying to reach a certain number. For \(d = \log_c{k}\), \(d\) signifies the number of times \(c\) must be increased to produce \(k\). It is crucial to remember:

• Base values: Typically greater than zero and not equal to one.
• Exponent values: Can be positive, negative, or zero depending on the relationship represented.

In converting between exponential and logarithmic forms, identifying the base and exponent allows accurate transformation, enabling deeper understanding of logarithmic expressions.