Problem 531 - Precalculus

Question

Use the definition of a logarithm to solve. \(-5 \log _{7}(10 n)=5\).

Step-by-Step Solution

Answer

The solution is \(n = \frac{1}{70}\).

Step 1: Isolate the Logarithm

First, we need to isolate the logarithmic expression. We start by dividing both sides of the equation by -5:\[-5 \log_{7}(10n) = 5\]Divide by -5 on both sides:\[\log_{7}(10n) = -1\]

Step 2: Convert Logarithmic Form to Exponential Form

Use the definition of logarithms to convert the equation to exponential form. The equation \( \log_{7}(10n) = -1 \) means the base 7 raised to the power of -1 equals \(10n\):\[7^{-1} = 10n\]This simplifies to:\[\frac{1}{7} = 10n\]

Step 3: Solve for Variable \(n\)

Now we solve for \(n\) by dividing both sides by 10:\[\frac{1}{7} = 10n\]Becomes:\[n = \frac{1}{70}\]

Key Concepts

Logarithmic Form

When you encounter a logarithmic equation, it's crucial to understand what each part represents. A logarithmic form of an equation, like \(\log_{7}(10n) = -1\), asks the question: "To what power must the base 7 be raised, to result in the number \(10n\)?" Here:

• The base is 7.
• The argument is \(10n\).
• The equation states that when 7 is raised to an unknown power, the result will be \(10n\).

Logarithms are a way to express powers. They reveal the power that a base is raised to in order to obtain another number. Grasping this concept helps when translating between logarithmic and exponential forms.

Exponential Form

Once you have the logarithmic equation \(\log_{7}(10n) = -1\), the next step is to convert it to exponential form. In exponential form, you state the equation in terms of powers:

• The base (7) raised to the power (-1) equals the argument (\(10n\)).
• This translates to \(7^{-1} = 10n\).

In exponential form, \(7^{-1}\) effectively means \(\frac{1}{7}\). This transformation is essential as it simplifies the process of solving for unknowns. When both sides express the same value in terms of exponentiation, it becomes clearer how to solve for the variable. Expanding this, you then know \(\frac{1}{7} = 10n\), which is the foundation for isolating the variable.

Isolate the Logarithm

Before translating a logarithmic equation into exponential form, it is necessary to isolate the logarithmic part of the equation. For the original equation \(-5 \log_{7}(10n) = 5\), we must first simplify it:

• Divide both sides by -5 to get the logarithm by itself.
• After division, the equation becomes \(\log_{7}(10n) = -1\).

Isolating the logarithm simplifies the equation significantly. It makes the equivalence clear and sets up the conversion to exponential form. Understanding how to isolate helps in tackling various equations and pivoting to the most straightforward solution paths smoothly.