Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in logarithmic form. \(m^{-7}=n\)

Step-by-Step Solution

Answer

\(\log_m(n) = -7\)

Step 1: Understand the Exponential Equation

We are given the equation \(m^{-7}=n\), which is in an exponential form. The base is \(m\), the exponent is \(-7\), and the result is \(n\).

Step 2: Apply the Definition of Logarithms

The logarithmic form of an equation helps express the relationship between the base, the exponent, and the result in terms of logarithms. By definition, if \(a^b = c\), then \(\log_a(c) = b\). With base \(m\), exponent \(-7\), and result \(n\), we write: \(\log_m(n) = -7\).

Step 3: Rewrite the Given Equation

Convert the exponential equation \(m^{-7}=n\) into the equivalent logarithmic expression using the understanding from Step 2. Thus, the equation becomes \(\log_m(n) = -7\).

Key Concepts

Exponential Form

Exponential form is a way to express numbers through a base raised to a power, which is referred to as the exponent. In the expression \(a^b = c\), "\(a\)" is the base and "\(b\)" is the exponent. The whole expression means that \(a\) multiplied by itself \(b\) times equals \(c\).

For example, in the given exercise, we have \(m^{-7}=n\). Here, \(m\) is the base, \(-7\) is the exponent, and \(n\) is the result.

This demonstrates a concept where exponentiation involves not just raising a number to a power but also interpreting negative and fractional exponents which reciprocate the base and apply the exponentiation afterwards. Thus, the exponential form offers a compact way to express large numbers or fractions.

Base and Exponent

Understanding the base and the exponent is crucial when dealing with exponential form expressions.

• Base: The base is the number that is getting multiplied. In \(a^b = c\), it is the "\(a\)".
• Exponent: The exponent indicates how many times the base is used as a factor. In \(a^b = c\), \(b\) is this exponent.

In our example from the exercise, \(m\) serves as the base, and \(-7\) is the exponent.

When exponents are negative, such as \(-7\), it implies a division by the base raised to the positive of that exponent. Therefore, \(m^{-7} = \frac{1}{m^7}\). Understanding the roles and effects of the base and exponent helps decode these expressions and prepare them for further manipulations such as converting into logarithmic form.

Logarithm Definition

A logarithm answers the question: "To what power must the base be raised to produce a certain number?" It essentially reverse engineers the exponential form.

The definition states: If \(a^b = c\), then \(\log_a(c) = b\). Here, the logarithm (log) denotes the power \(b\), indicating the extent to which base \(a\) was raised to result in \(c\).

In our exercise, the exponential equation is \(m^{-7} = n\). Following the definition of a logarithm, we can rewrite this in logarithmic form as \(\log_m(n) = -7\).

This tells us the exponent \(-7\) is necessary for base \(m\) to yield \(n\).

Logarithms act as powerful tools in simplifying equations, solving exponential equations, and understanding the growth patterns in different fields like computing, engineering, and natural sciences.