Problem 262 - Calculus Volume 1

Question

For the following exercises, write the equation in equivalent logarithmic form. $$ b^{3}=45 $$

Step-by-Step Solution

Answer

The equivalent logarithmic form is \( 3 = \log_b{45} \).

Step 1: Identify the Exponential Form

The given equation is an exponential form: \( b^3 = 45 \). In this equation, \( b \) is the base, \( 3 \) is the exponent, and \( 45 \) is the result.

Step 2: Convert to Logarithmic Form

To convert from exponential form to logarithmic form, we use the relationship \( b^y = x \) is equivalent to \( y = \log_b{x} \). For our equation, \( b^3 = 45 \), so the logarithmic form is \( 3 = \log_b{45} \).

Key Concepts

Exponential Form

Exponential form is a way of expressing numbers or equations where a base number is raised to a certain power, known as the exponent. In mathematical terms, it can be written as \(b^y = x\), where \(b\) is the base, \(y\) is the exponent, and \(x\) is the result or the power.Understanding exponential form is crucial for simplifying expressions and solving equations. It allows big numbers to be expressed in a more manageable form.

• The base \(b\) in exponentials must be a positive number, except zero.
• The exponent \(y\) tells us how many times to multiply the base by itself.
• The result \(x\) is what you get after performing the multiplication.

For example, in the equation \(b^3 = 45\), the base is \(b\), the exponent \(3\), indicating \(b\) is multiplied by itself three times, producing 45 as the result.

Logarithmic Form

Logarithmic form is another way to express exponential relationships. This form allows us to solve equations more easily by rewriting the exponential form as \(y = \log_b{x}\). Here, \(\log_b{x}\) represents the logarithm of \(x\) with base \(b\), equivalent to saying \(b^y = x\).Converting from an exponential to logarithmic form is helpful in solving questions that involve finding the exponent. It mostly focuses on finding unknowns in exponential scenarios by setting them in log equations.

• \(b\) in the expression \(\log_b{45}\) remains the base of the logarithm.
• The exponent in the exponential equation becomes the result in the logarithmic form.
• The result from the exponential equation, \(x\), is the number you are logging with base \(b\).

In the exercise, the conversion \(b^3 = 45\) becomes \(3 = \log_b{45}\). This maintains the mathematical relationship in a different perspective, aiding in different uses such as solving for the base or exponent.

Equivalent Expressions

Equivalent expressions are essential in mathematics as they allow the same expression or equation to be represented in various forms. This concept applies to exponential and logarithmic equations, where one equation can be expressed in different forms, yet represents the same value or relationship.Understanding equivalent expressions means realizing that changing the form of an equation doesn’t change its inherent truth or relationship. When you convert an exponential equation to a logarithmic one, you are simply representing the same exponential relationship via logarithms.

• Both exponential \(b^y = x\) and logarithmic \(y = \log_b{x}\) forms represent identical relationships.
• Switching between forms is typically done to make the task at hand, such as solving for unknowns, more straightforward.
• Using equivalent expressions allows flexibility in problem-solving and can simplify complex calculations.

For the equation \(b^3 = 45\), converting this to a logarithmic form \(3 = \log_b{45}\) illustrates how the same mathematical principle is at play in different but equivalent ways.