Problem 13 - Precalculus Mathematics for Calculus

Question

Express the equation in logarithmic form. (a) \(5^{3}=125\) (b) \(10^{-4}=0.0001\)

Step-by-Step Solution

Answer

(a) \( \log_5(125) = 3 \); (b) \( \log_{10}(0.0001) = -4 \).

Step 1: Understand the Relationship

Logarithmic form is essentially the inverse of exponential form. The equation in exponential form is given as \( b^x = y \), which in logarithmic form is expressed as \( \log_b(y) = x \). Here, \( b \) is the base, \( x \) is the exponent, and \( y \) is the result.

Step 2: Convert First Equation

For the equation \( 5^3 = 125 \), identify \( b = 5 \), \( x = 3 \), and \( y = 125 \). Applying the logarithmic form, we write: \( \log_5(125) = 3 \).

Step 3: Convert Second Equation

For the equation \( 10^{-4} = 0.0001 \), identify \( b = 10 \), \( x = -4 \), and \( y = 0.0001 \). Applying the logarithmic form, we write: \( \log_{10}(0.0001) = -4 \).

Key Concepts

Exponential Form

When we talk about the exponential form of an equation, we mean an expression where a number, called the base, is raised to a power or exponent to produce another number. Here’s how it looks:

• The base is the number that is being multiplied.
• The exponent tells you how many times to multiply the base by itself.
• The result is the outcome of this multiplication.

For example, in the expression \(5^3\), 5 is the base, and 3 is the exponent. It's read as 5 raised to the power of 3, which equals 125. This is known as the exponential form. This form is useful because it provides a simple and concise way to represent repeated multiplication.

Inverse Relationship

The inverse relationship in mathematics is about swapping the roles of input and output in a function. For exponential and logarithmic forms, their relationship is reciprocal. When you convert an exponential equation into a logarithmic one, you're essentially finding the exponent in the exponential equation.

• In exponential form: \(b^x = y\).
• In logarithmic form: \(\log_b(y) = x\).

Here, if you're given the exponential form and need to find what the exponent is for a given base and result, you use the concept of logarithms. This is because finding a logarithm is essentially performing the inverse operation of exponentiation. It's like asking the question, "To what power must the base be raised to give this result?" Understanding this inverse relationship is key to converting between exponential and logarithmic forms.

Base of a Logarithm

The base in a logarithmic equation is the same as in its corresponding exponential form. It's the number that you repeatedly multiply by itself. When writing in logarithmic form, the base is written as a small subscript to the word "log." For instance, in the logarithmic expression \(\log_5(125) = 3\):

• The base is 5.
• This base is being used to find what power of 5 equals 125.

A crucial point to remember is that the base of the logarithm must always be a positive number and cannot be 1, as these values would make the operation meaningless or lead to undefined results. The base effectively shapes the way numbers are transformed in both exponential and logarithmic expressions.

Exponents

Exponents are a mathematical way to express how many times a number, the base, is used in a multiplication. They are part of exponential expressions such as \(b^x\), where this little number, x, on top of the base is the exponent.Exponents have several important properties:

• They simplify the representation of large numbers.
• They define the power to which the base is raised.
• They allow for calculations involving growth and decay, among other applications.

In an equation like \(5^3 = 125\), 3 is the exponent, indicating that 5 should be multiplied by itself three times. Understanding exponents is vital because converting between exponential and logarithmic forms involves identifying what the exponent is for a given base and result.