Problem 261 - Calculus Volume 1

Question

Write the equation in equivalent logarithmic form. \(\quad 9^{y}=150\)

Step-by-Step Solution

Answer

\(y = \log_9{150}\)

Step 1: Identify the logarithmic equivalence

In order to convert the exponential form \(9^y = 150\) into logarithmic form, identify the parts of the equation: the base \(b = 9\), the exponent \(y\), and the result or power \(150\).

Step 2: Apply the definition of a logarithm

Using the definition of a logarithm, which states that if \(b^y = x\), then \(\log_b{x} = y\), rewrite the equation \(9^y = 150\) as a logarithm. The base \(b\) is \(9\), the exponent \(y\) is what the logarithm equals to, and the result \(x\) is \(150\).

Step 3: Write the equivalent logarithmic equation

Following the definition, write the equivalent logarithmic equation: \(y = \log_9{150}\). This represents the same relationship as \(9^y = 150\), but in logarithmic form.

Key Concepts

Exponential Form

In mathematics, the exponential form of an equation like \(9^{y} = 150\) expresses a power relationship between a base and an exponent. Here, "9" is the base, "\(y\)" is the exponent, and "150" is the result or power.
When you see an equation in this form, it simply means that the base number is multiplied by itself a certain number of times, as indicated by the exponent. In our case, we're looking for how many times 9 has to multiply itself to reach 150.

• Base: The number that is being multiplied. In this context, it's 9.
• Exponent: The number that tells you how many times to multiply the base. It's represented as \(y\) here.
• Result/Power: The outcome of the base raised to the exponent, which is 150.

Understanding this allows us to interpret and transform the equation into other forms, such as the logarithmic form, to solve it or understand it better.

Logarithmic Form

The logarithmic form is the counterpart to the exponential form, allowing us to express the relationship in a different way. For the equation \(9^{y} = 150\), its logarithmic form is \(y = \log_{9}150\).
This form answers the question: "To what power must the base, in this case 9, be raised to obtain 150?" Using logs, we essentially swap places of the exponent and result to make the power the subject.

• The base remains the same: 9.
• The result becomes the inside of the log function: 150.
• The exponent becomes what the logarithm equals: \(y\).

This transformation helps us because sometimes it's more natural or more straightforward to solve problems using logs, particularly if you're solving for the exponent.

Definition of Logarithm

The definition of a logarithm provides the bridge between exponential and logarithmic equations. It states that if an equation in exponential form is \(b^y = x\), then the logarithmic form is \(\log_b{x} = y\).
Logarithms essentially serve the purpose of undoing the exponentiation operation. While exponential growth is about multiplying, logarithms tell us how many of those multiplications we need to get to a certain number.
Here’s how we break it down:

• **Base**: This is the same for both exponential and logarithmic forms. It's the number being raised to a power (in \(\log_9{150}\), the base is 9).
• **Exponent/Logarithm Result**: This is what we aim to find with logarithms (\(y\) in this case, the value of the log).
• **Application**: By applying the definition of logarithm, we make finding that exponent simpler, particularly for non-standard bases or when exact answers are necessary but not directly evident.

This understanding allows us to convert back and forth between forms, which is crucial in both pure and applied mathematics.