Problem 254 - Calculus Volume 1

Question

For the following exercises, write the equation in equivalent logarithmic form. $$ 2^{3}=8 $$

Step-by-Step Solution

Answer

\(\log_2{8} = 3\)

Step 1: Identify Components of the Exponential Equation

The given exponential equation is \(2^3 = 8\). Here, the base is 2, the exponent is 3, and the result is 8.

Step 2: Understand the Logarithmic Form

In logarithmic form, an equation \(b^x = y\) can be rewritten as \(\log_b{y} = x\). The base \(b\) of the exponential becomes the base of the logarithm, \(x\) is the logarithm itself, and \(y\) is the value obtained from the exponential equation.

Step 3: Substitute Values into Logarithmic Form

From the exponential equation \(2^3 = 8\), identify the equivalent components: the base \(b = 2\), the result \(y = 8\), and the exponent \(x = 3\). Substitute these values into the logarithmic form \(\log_b{y} = x\) to get \(\log_2{8} = 3\).

Step 4: Write the Final Equation

After substituting the values, the equivalent logarithmic form of the given exponential equation is \(\log_2{8} = 3\). This states that the logarithm of 8 with base 2 is 3.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where a fixed number, called the base, is raised to a variable power. In general terms, an exponential equation has the form \(b^x = y\), where \(b\) is the base, \(x\) is the exponent, and \(y\) is the outcome or result. These equations can seem straightforward but hold powerful implications in different fields such as science and finance.

When tackling exponential equations, it's crucial to

• Understand the components: base, exponent, and result.
• Be aware of how changes in the exponent affect the result.
• Recognize that these equations often describe exponential growth or decay processes.

For instance, in the equation \(2^3 = 8\), the number 2 is the base, 3 is the exponent, and the outcome is 8.

Logarithmic Form

Transforming an exponential equation into a logarithmic form is a key technique in mathematics. This conversion makes it easier to solve equations involving exponential parts by working with logarithms.

In the logarithmic form, an equation like \(b^x = y\) is expressed as \(\log_b y = x\). This means that the exponent becomes the result of the logarithmic function, showing how many times the base must be multiplied by itself to reach the result \(y\).

The conversion process involves

• Identifying the base of the original equation as the base of the logarithm.
• Replacing the result of the exponential equation with what the logarithm equates to.
• Setting the exponent of the exponential equation as the outcome of the logarithmic equation.

For example, converting \(2^3 = 8\) to its logarithmic form gives \(\log_2 8 = 3\). This tells us the power to which 2 must be raised to yield 8 is 3.

Base and Exponent Relationship

The relationship between the base and the exponent is fundamental to understanding both exponential and logarithmic equations. The base is the number that gets multiplied, while the exponent indicates how many times the base is used in the multiplication.

Consider the base-exponent relationship in the example \(2^3 = 8\):

• The base (2) is used as a factor three times (since the exponent is 3).
• This results in the multiplication \(2 \times 2 \times 2 = 8\).
• The power to which a base is raised can drastically change the result, highlighting the importance of the exponent.

When interpreting this relationship, it's crucial to grasp that the exponential expression indicates repeated multiplication of the base, which is why it's succinctly represented with an exponent. This foundational concept is pivotal in advancing to logarithms, where the exponent morphs into the logarithmic result as seen in \(\log_2 8 = 3\).