Problem 31 - College Algebra with Corequisite Support

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 solution to the equation is \(n = 49\).

Step 1: Isolate the logarithm

To solve the equation \(5 \log_{7} n = 10\), first divide both sides of the equation by 5 to isolate the logarithmic expression. This gives us: \(\log_{7} n = \frac{10}{5}\), which simplifies to \(\log_{7} n = 2\).

Step 2: Rewrite in exponential form

Using the definition of a logarithm, \(\log_{b} x = y\) implies \(x = b^y\). Apply this definition to \(\log_{7} n = 2\), giving \(n = 7^2\).

Step 3: Compute the Exponent

Calculate \(7^2\) to find the value of \(n\). \(7^2 = 49\), so \(n = 49\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are vital in representing processes that grow or decay at a constant rate, like population growth, radioactive decay, and compound interest.

Characteristics of exponential functions include:

• A constant base, which is a positive real number other than 1.
• An exponent, which may be any real number, commonly a variable like 'x'.
• The general form is expressed as: \(f(x) = a \cdot b^x\), where 'a' is the initial amount, 'b' is the base, and 'x' is the exponent.

In an exponential function, increasing the exponent results in the function increasing or decreasing rapidly, depending on whether the base is greater than 1 (growth) or between 0 and 1 (decay).

Understanding exponential functions helps in solving problems like converting logarithmic equations into exponential form, which is a key step in solving logarithmic equations.

Equations

Equations are mathematical statements that assert the equality of two expressions. An equation consists of two expressions separated by an equal sign "=". Solving an equation involves finding the values of the variables that make the equation true.

There are various types of equations, including linear, quadratic, and exponential equations. Each type may require different methods for solving:

• Linear equations involve constants and variables raised to the first power and are solved using simple algebraic techniques.
• Quadratic equations include variables raised to the second power and often require factoring or using the quadratic formula for solutions.
• Exponential equations involve variables in the exponent and typically require the use of logarithms for solving.

A key part of solving any equation is understanding the properties of equality and using operations that keep the equation balanced. In logarithmic equations, one typically isolates the logarithmic expression before rewriting it in its exponential form to find the solution.

Solving Logarithmic Equations

Solving logarithmic equations involves several steps to isolate and simplify the logarithmic expression to find the value of the variable of interest.

Let's break down the process:

• Isolate the logarithm: Begin by rearranging the equation to isolate the logarithmic expression, much like our initial equation with \(5 \log_{7} n = 10\), which can be simplified to \(\log_{7} n = 2\).
• Exponentiate to remove the log: Convert the logarithmic expression into an exponential form. Using the property \(\log_{b} x = y\) implies \(x = b^y\), our equation becomes \(n = 7^2\).
• Calculate the result: Perform the necessary calculations to solve for the variable, resulting in \(7^2 = 49\), so \(n = 49\).

This methodical approach aids in understanding how logarithms and exponents are interrelated and provides a systematic way to solve logarithmic equations, ensuring accuracy and clarity in reaching the solution.