Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3$$4^{4}=256\)

Step-by-Step Solution

Answer

The logarithmic form of the equation \(4^{4}=256\) is \(\log _{4} 256=4\).

Step 1: Identify the parts of the exponential equation

In the equation \(4^{4}=256\), 4 is the base of the exponent, 4 is the exponent, and 256 is the result of the exponential operation.

Step 2: Convert exponential form to logarithmic form

Using the formula \(\log _{a} y=x\), where 'a' is the base, 'y' is the result of the operation, and 'x' is the exponent, we can convert \(4^{4}=256\) into logarithmic form as \(\log _{4} 256=4\)

Key Concepts

Definition of a Logarithm

Understanding the definition of a logarithm is foundational to working with various mathematical problems, especially when dealing with exponential equations. A logarithm can be seen as the opposite of exponentiation; it answers the question, 'To what power must the base be raised to obtain a certain number?'. For example, if you have an exponential equation like \( b^x = y \), in logarithmic terms it is represented as \( \log_b(y) = x \). Here, 'b' is the base, 'y' is the result you end up with, and 'x' is the exponent to which the base has been raised. This relationship is the core concept of logarithms and is crucial for not only understanding but also for solving different types of algebraic and geometric problems.

Using the language of logarithms, we are essentially unpacking the exponent to reveal the power needed for a base to result in a certain number. Now let's dive into exploring how they are actually used in equations.

Exponential Equations

Exponential equations feature variables in the exponent and take the form \( a^x = b \), where 'a' is the base, 'x' is the exponent, and 'b' is the constant which is the result of the base raised to the power of 'x'. These equations are omnipresent across various fields of study, from compound interest calculations in finance to radioactive decay in physics. Since the variable is an exponent, the methods used to solve exponential equations are different from those applied to solving linear or quadratic equations.

One technique for solving such equations is by utilizing the properties of logarithms. This is because logarithms can help us transform the exponent into a form that allows us to isolate the variable. An important property to remember is that the base of the exponential equation is also the base of the logarithm when we convert from exponential form to logarithmic form. This transition from exponential to logarithmic form is a significant step in solving these equations, as we will soon illustrate.

Converting Exponentials to Logarithms

Converting an exponential equation to its logarithmic form is a critical skill for solving various problems in algebra and calculus. This process involves rewriting the exponential expression as a logarithm. As we've seen in the step-by-step solution to the exercise, the transformation uses a formula, \( \log_b(y) = x \), which allows us to see the components of an exponential equation in a new light.

Let's apply this to the given problem: \( 4^4 = 256 \). This equation in logarithmic form translates to \( \log_4(256) = 4 \). This conversion is essential because it lets us solve for the exponent, which may not always be as straightforward as in this example. Moreover, when working with more complex equations, understanding this conversion gives students the tools to handle situations where exponents aren't easily compared or deduced.

To put it in practice, remember this guideline: identify the base, exponent, and the result of the operation in the exponential form. Then, place the base in the subscript of the logarithm, the result as the argument of the logarithm, and the exponent as the outcome of the logarithmic equation. This conversion procedure is powerful as it provides a bridge to solving the equations by extracting the exponent from the equation's original exponential form.