Problem 31 - College Algebra

Question

For the following exercises, use the definition of a logarithm to solve the equation. $$ 5 \log _{7}(n)=10 $$

Step-by-Step Solution

Answer

The value of \(n\) is 49.

Step 1: Understand the logarithmic equation

The given equation is \(5 \log_{7}(n) = 10\). This implies that \(\log_{7}(n)\) is multiplied by 5.

Step 2: Isolate the logarithm

To isolate \(\log_{7}(n)\), divide both sides of the equation by 5. This gives us \(\log_{7}(n) = 2\).

Step 3: Use the definition of the logarithm

The definition of a logarithm is that if \(\log_{b}(a) = c\), then \(b^c = a\). Apply this definition to \(\log_{7}(n) = 2\), which means that \(7^2 = n\).

Step 4: Calculate the value of n

Since \(7^2 = n\), calculate the value of \(n\). Thus, \(n = 49\).

Key Concepts

Solving Logarithmic Equations

When solving logarithmic equations, the goal is to isolate the logarithmic expression before converting it into an exponential form. Let's break this down using our example equation, \(5 \log_{7}(n) = 10\). Our first step in tackling this equation is to get \(\log_{7}(n)\) by itself. This means we need to move any coefficients or additional terms that are either multiplying or adding to the logarithm.
In this particular problem, we have the term \(5\) multiplying \(\log_{7}(n)\). To isolate \(\log_{7}(n)\), we divide both sides of the equation by \(5\). This simplifies our equation to:\[\log_{7}(n) = 2\]
Now that we've isolated the logarithmic part, we can move on to the next step of solving it. Remember, isolating the logarithm is crucial because it allows us to apply logarithmic properties effectively.

Logarithmic Properties

Logarithmic properties are a set of rules that allow us to simplify and manipulate logarithm expressions. The essence of logarithms lies in understanding how to switch between logarithmic and exponential forms using these properties.
One of the fundamental properties of logarithms is the definition itself, which is represented as if \(\log_{b}(a) = c\), then \(b^c = a\). This property is key in converting a log equation into an exponential one.
In our problem, once we have \(\log_{7}(n) = 2\), we can use this property to transform the equation from logarithmic form to exponential form:

• The base \(b\) is \(7\)
• The exponent \(c\) is \(2\)
• The value \(a\) is \(n\)

Using this information, we rewrite the equation as \(7^2 = n\). Understanding and applying these properties makes it easier to solve an array of logarithmic equations.

Exponentiation

Exponentiation is a mathematical operation that involves raising a base number to the power of an exponent. It is the inverse operation of finding a logarithm and comes into play once we have isolated our logarithmic expression and applied the relevant properties.
Let's consider the equation \(7^2 = n\). Here, \(7\) is our base number, and \(2\) is the exponent. Exponentiation simplifies to calculating \(7 \times 7\), which equals \(49\).
This gives us the value \(n = 49\), completing the solution to our original logarithmic equation. By understanding the relationship between logarithms and exponents, you can easily swap between these forms, allowing for smooth and efficient problem-solving. Remember, mastering this operation is crucial, as it frequently appears in various areas of math and science.