Question

Rewrite the equation in exponential form. (See Example 1.) \(\log _4 4=1\)

Short answer

The exponential form of the given logarithmic equation \(\log _4 4=1\) is \(4^1 = 4\).

Step-by-step solution

Understand the given logarithmic equation.

The given equation is \(\log _4 4=1\). It states that the logarithm to the base 4 of 4 is equal to 1.

Rewrite in exponential form.

The exponential form of a logarithmic equation \(\log_b a = c\) is \(b^c = a\). Substitute \(b=4\), \(a=4\), and \(c=1\) into the exponential form, which gives \(4^1 = 4\).

Key concepts

Logarithms

To understand how to convert logarithmic to exponential form, let's first dive into the concept of logarithms. A logarithm answers the question: 'To what power must we raise a base, so that we obtain a certain number?' The general format for a logarithm is \( \log_b a = c \) where \(b\) is the base, \(a\) is the result we want, and \(c\) is the power we raise the base to obtain \(a\). In simple terms, if we say \( \log _4 16 = 2 \) we mean '4 raised to what power gives us 16?' The answer is '2'. Understanding this concept is crucial to working with logarithms in algebra.

Exponential Equations

Exponential equations are the inverse of logarithmic equations. They are written in the form \( b^c = a \) where \(b\) is the base, \(c\) is the exponent, and \(a\) is the result of the base raised to the power. These equations pop up in various areas of mathematics and science, particularly in studies involving growth and decay, such as population models or radioactive decay. For example, if we have an exponential equation like \( 2^3 = 8 \) it shows that the base 2 raised to the third power equals 8. Switching between logarithmic and exponential forms is a key skill in algebra and helps in solving complex equations more easily.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and include numbers, variables, and arithmetic operations. These expressions represent quantities without fixed values, often unknowns that we aim to solve for. Algebraic expressions can take many forms, such as polynomials, rational expressions, and yes, even logarithmic expressions. For instance, \( x^2 - 5x + 6 \) is a polynomial expression, while \( \log_b x \) is a logarithmic expression. When you grasp how to manipulate these expressions—like rewriting logarithms in exponential form—you’ve got a powerful toolset for solving algebra problems.

Mathematical Transformations

Mathematical transformations involve changing the appearance or form of a mathematical entity while retaining its intrinsic properties. In the context of algebra, transforming a logarithmic equation to exponential form is a kind of transformation. This doesn't alter the equation's meaning; it just presents the relationship between elements differently. Transformations are used for simplification, to obtain different perspectives of an equation, or to make an equation solvable using available methods. The process of learning transformations goes hand-in-hand with understanding the fundamental operations of algebraic manipulation.