Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in exponential form. \(\log (v)=t\)

Step-by-Step Solution

Answer

The exponential form is \( v = 10^t \).

Step 1: Identify the Logarithmic Equation

The given equation is a logarithmic equation: \( \log (v) = t \). This tells us that the logarithm of \( v \) is equal to \( t \).

Step 2: Understand the Exponential Form

The exponential form corresponding to a logarithm \( \log_b(a) = c \) is \( a = b^c \). Here, \( b \) is the base for the logarithm. In this problem, the base is not explicitly mentioned, which means it is assumed to be 10 (common logarithm).

Step 3: Rewrite in Exponential Form

Since we have \( \log (v) = t \), and it is assumed to be a common logarithm, this is equivalent to saying that \( v = 10^t \). We rewrote the equation from logarithmic to exponential form by using the base 10.

Key Concepts

Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm with a variable inside it. Logarithms are the inverse operations of exponential functions. In these equations, you often see a format such as \( \log_b(a) = c \). This suggests that the variable \( a \) is the value you raise the base \( b \) to in order to get the result \( c \).
Logarithmic equations are useful for solving problems where the variable appears as an exponent. By rewriting them in exponential form, you can often find solutions more directly.
Understanding and identifying a logarithmic equation is crucial because it allows us to use the properties of logarithms and the relationship between logarithms and exponents effectively.

Common Logarithm

The common logarithm is a logarithm that has a base of 10. This is the most frequently used base in logarithms and hence does not need to be explicitly mentioned. In mathematics, when you see a logarithm like \( \log(v) \), it typically implies \( \log_{10}(v) \).
The common logarithm has many applications, especially in scientific calculations, because it is based on our decimal number system. It is also the logarithm you are most likely to encounter in everyday calculations and scientific data.
By default, when the base of the logarithm is absent, it is safer to assume it to be 10 unless specified otherwise. This small assumption can make a big difference in how you interpret and solve logarithmic equations.

Converting Equations

Converting logarithmic equations into exponential form is a valuable skill because it allows simpler manipulation of the expression. The process is quite straightforward: if you have \( \log_b(a) = c \), then you can rewrite it in exponential form as \( a = b^c \).
Converting to exponential form helps to visualize the relationship between the numbers more clearly. In our example, \( \log(v) = t \) implies \( v = 10^t \), where the process of conversion made use of the fact that the base was 10, a common logarithm.
Such conversions are not only useful in solving equations but also essential in applications ranging from data transformations in engineering to statistical modeling. Always ensure the base is correctly identified before converting.