Question

If \(\log _{3}(m-n)=0\) and \(\log _{3}(m+n)=3\), determine the values of \(m\) and \(n\).

Short answer

The values are \(m = 14\) and \(n = 13\).

Step-by-step solution

Understand the Logarithmic Equations

Given the equations: 1. \(\log _{3}(m-n)=0\) 2. \(\log _{3}(m+n)=3\)

Convert Logarithmic Form to Exponential Form

Use the definition of logarithms to convert each equation to its exponential form. For \(\log _{3}(m-n)=0\), we get \(m-n = 3^0 = 1\). For \(\log _{3}(m+n)=3\), we get \(m+n = 3^3 = 27\).

Solve the System of Equations

We have two equations now: 1. \(m - n = 1\) 2. \(m + n = 27\). To solve these, add the two equations: \( (m - n) + (m + n) = 1 + 27\), which simplifies to \(2m = 28\) or \(m = 14\).

Find the Value of n

Substitute \(m = 14\) back into the first equation: \(14 - n = 1\), which simplifies to \(n = 13\).

Key concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions that help us understand how to compare different powers of a fixed base. A logarithm finds the exponent to which a base number must be raised to produce another number. In this exercise, we are given two logarithmic equations involving a base of 3. These equations are: \( \log_3(m-n) = 0 \) and \( \log_3(m+n) = 3 \). Since both equations have the same base, we can use the properties of logarithms to solve for the variables involved. Logarithmic functions are the inverse operations of exponential functions, meaning they can convert multiplicative relationships into additive ones. This property is crucial as it allows us to transform the logarithmic equations into a more familiar form.

Exponential Form Conversion

One of the key steps in solving logarithmic equations is converting them into their exponential form. This turns the logarithmic expression into an equation where the exponent can be easily identified. Using the logarithmic definitions from the exercise, the equation \( \log_3(m-n) = 0 \) converts to: \[ m - n = 3^0 = 1 \]. This tells us that the difference between m and n is 1. Similarly, converting \( \log_3(m+n) = 3 \) to exponential form gives us: \[ m + n = 3^3 = 27 \]. Here, it shows that the sum of m and n is 27. Converting logarithms to exponential form simplifies the expressions and makes it easier to work with them in subsequent steps.

System of Equations

Once logarithmic equations are converted into exponential form, they often result in a system of linear equations. In this case, we have two equations: \( m - n = 1 \) and \( m + n = 27 \). To solve for m and n, we need to find values that satisfy both equations simultaneously. Adding the two equations together eliminates n: \( (m - n) + (m + n) = 1 + 27 \), which simplifies to: \[ 2m = 28 \]. Solving for m gives: \[ m = 14 \]. Substituting m back into one of the original equations, \( 14 - n = 1 \), we solve for n: \[ n = 13 \]. Therefore, the solution to the system is m = 14 and n = 13. Understanding how to solve systems of equations is crucial as it is a common method used in algebra to find the values of unknown variables.