Problem 53 - Algebra 2

Question

Write each equation in exponential form. $$ \log _{2} 128=7 $$

Step-by-Step Solution

Answer

The equivalent exponential form of the given logarithmic equation is \(2^7 = 128\).

Step 1: Identify the base, argument, and logarithm of the logarithmic equation

In the given equation, \(\log_2 128 = 7\), the base \(b\) is 2, the argument \(a\) is 128, and the logarithm \(n\) is 7.

Step 2: Write the equivalent exponential equation

Using the transformation \(b^n = a\), substituting the identified values from the logarithmic equation gives \(2^7 = 128\).

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics which are essentially the inverse operation of exponentiation. They help us solve equations where the unknown appears as the exponent of a number. The notation \(\log_b a\) refers to the logarithm of \(a\) with base \(b\). This means that there is a power, usually represented by \(n\), such that \(b^n = a\). Logarithms answer the question: "To what power should the base \(b\) be raised to obtain the number \(a\)?" understanding this is crucial for solving many types of equations, including logarithmic and exponential ones. With logarithms, you convert multiplicative processes into additive ones. This property makes them quite useful in areas ranging from statistics to computer science.

When working with logarithms, make sure to familiarize yourself with the laws that govern them, like the product law, the quotient law, and the power law, as these can simplify many calculations.

Base and Argument

In a logarithmic expression like \(\log_b a\), \(b\) is referred to as the "base" and \(a\) is called the "argument." The base \(b\) is the number that is raised to a power to obtain the argument \(a\). In our example \(\log_{2} 128 = 7\), the base is 2. This means we are interested in understanding how many times you multiply 2 by itself to reach 128.

The argument is the number for which the logarithm is being found, here it is 128. It represents the result of multiplying the base repeatedly by itself as specified in the logarithm. Comprehending the relationship between the base and the argument is key to correctly converting between logarithmic and exponential forms.

• The base is always the "foundation" you build your power from.
• The argument is the "structure" resulting from that power.

It's important to note that logarithms are only defined for positive real arguments and bases that are positive and not equal to one.

Exponential Equations

Exponential equations are equations in which the unknown appears as an exponent. These equations commonly take forms like \(b^x = a\). Solving exponential equations usually involves rewriting the equation in a form that makes the relationship more transparent; often, this involves switching to logarithmic form if working with complicated powers.

In our example, the logarithmic equation \(\log_{2} 128 = 7\) was converted to an exponential form \(2^7 = 128\). This relationship is direct and helps us understand the mechanics of these transformations. Exponential equations frequently appear in real-world scenarios, such as calculating compound interest or growth in populations. Understanding these will provide you with tools to address these problems efficiently.

• Recognize the base and exponents to make conversions easier.
• Use logarithmic transformations to solve for unknown exponents.

The transformation between logarithmic and exponential form is a powerful technique, allowing us to switch perspectives and solve problems in various mathematical and practical contexts.