Problem 7 - College Algebra

Question

For the following exercises, rewrite each equation in exponential form. $$\log _{\mathrm{a}}(b)=c$$

Step-by-Step Solution

Answer

The exponential form is \( a^{c} = b \).

Step 1: Understanding the Logarithm

The given equation is in the form of a logarithm, written as \( \log_{a}(b) = c \). This expression means that the base \( a \) raised to the power of \( c \) equals \( b \). In other words, the logarithm gives us the exponent \( c \) needed for the base \( a \) to result in \( b \).

Step 2: Rewriting in Exponential Form

To convert the logarithmic equation \( \log_{a}(b) = c \) into exponential form, we apply the definition of a logarithm. The base of the logarithm becomes the base of the exponent: \( a \). The exponent is the solution of the original logarithm, which is \( c \). Thus, the converted equation will be \( a^{c} = b \).

Key Concepts

Logarithms

Logarithms are mathematical tools that help us solve problems involving exponential growth or decay. They might sound intimidating at first, but they simply answer the question: "To what power must we raise this base to get that number?" For instance, in the expression \( \log_{a}(b) = c \), the logarithm helps us find out the power \( c \) to which the base \( a \) must be raised to yield \( b \).
Logarithms have a few key properties that are helpful:

• The base of the logarithm, denoted as \( a \), is the number that gets raised to the power \( c \).
• The "argument" of the logarithm, \( b \), is the result we achieve after raising \( a \) to the power \( c \).
• The logarithm's "output" or result is \( c \), the exponent itself.

By understanding these basic elements, you can tackle logarithmic expressions with confidence and ease.

Exponents

Exponents are the shorthand way mathematicians use to express repeated multiplication of the same number. In the equation below, \( a^{c} = b \), the letter \( a \) represents the "base", while \( c \) is the "exponent" or "power" that tells us how many times to multiply the base by itself.
Here's how it works: if \( a = 2 \) and \( c = 3 \), then \( a^{c} = 2 \times 2 \times 2 = 8 \). Exponents provide a convenient way to handle large numbers or expressions through multiplication.
There are important properties of exponents that are worth noting:

• Any base raised to the zero power is 1, i.e. \( a^0 = 1 \).
• Negative exponents indicate reciprocal values, i.e. \( a^{-c} = \frac{1}{a^{c}} \).
• Fractional exponents correspond to roots, so \( a^{\frac{1}{c}} = \sqrt[c]{a} \).

Understanding these properties will make working with exponents much easier.

Mathematical Expressions

Mathematical expressions involve a variety of symbols and numbers to represent values or operations. In our context, an expression like \( \log_{a}(b) \) or \( a^{c} = b \) is a concise way to communicate complex mathematical ideas.
When dealing with such expressions, it's important to grasp the roles of each part involved:

• The base \( a \) is the foundation of either the logarithm or exponent.
• The number \( b \) is the outcome that results from applying the base and exponent.
• The exponent, often derived from solving a logarithmic expression, indicates the number of times the base is used in multiplication.

Mathematical expressions allow us to efficiently and clearly describe mathematical concepts and relationships, making them vital to both understanding and solving problems in mathematics.