Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, solve each equation for \(x\). \(\log _{3}(3 x)-\log _{3}(6)=\log _{3}(77)\)

Step-by-Step Solution

Answer

The solution is \(x = 154\).

Step 1: Apply Logarithm Property

We start by using the properties of logarithms. The equation is: \(\log_{3}(3x) - \log_{3}(6) = \log_{3}(77)\).The difference of two logarithms can be expressed as the logarithm of a quotient: \(\log_{3}\left(\frac{3x}{6}\right) = \log_{3}(77)\).

Step 2: Simplify the Quotient

Now simplify the expression inside the logarithm on the left side:\(\log_{3}\left(\frac{3x}{6}\right)\) simplifies to \(\log_{3}\left(\frac{x}{2}\right)\).So, the equation becomes: \(\log_{3}\left(\frac{x}{2}\right) = \log_{3}(77)\).

Step 3: Equate the Arguments

Since the bases of the logarithms are the same, we can equate the arguments:\(\frac{x}{2} = 77\).

Step 4: Solve for x

To solve for \(x\), multiply both sides by 2:\(x = 77 \times 2\).Thus, \(x = 154\).

Key Concepts

Understanding Logarithm Properties

To solve equations involving logarithms, it's crucial to understand the basic properties they possess. Logarithms convert multiplicative relationships into additive ones, and this feature is what makes them so powerful. In our equation, we have a case where we can use the property that allows us to express the subtraction of two logarithms as a single logarithm of a quotient:

• If you have an equation of the form \[\log_b(A) - \log_b(B) = \log_b(C)\], this can be rewritten as \[\log_b\left(\frac{A}{B}\right) = \log_b(C)\]
• This transformation is based on one of the key logarithm properties: \[\log_b\left(\frac{A}{B}\right) = \log_b(A) - \log_b(B)\]

With these properties in mind, we can manipulate and simplify the logarithmic expressions effectively, making it easier to solve the equation.

Simplifying Expressions Inside Logarithms

Once we've applied the logarithm property to convert the equation into a single logarithm, our next step is simplifying the expression within this logarithm. In our example, we transformed the equation to:\[\log_3\left(\frac{3x}{6}\right) = \log_3(77)\]By dividing inside the logarithm, we simplify \(\frac{3x}{6}\) to \(\frac{x}{2}\). This is a vital step because it often simplifies the problem dramatically, reducing complex expressions into simpler forms:

• Simplifying \[\log_3\left(\frac{3x}{6}\right)\] provides us \[\log_3\left(\frac{x}{2}\right)\]
• This simplification helps in making the equation more approachable for solving

Remember that expressing terms inside logarithms in their simplest form is a common theme in solving logarithmic equations.

Equating the Arguments of Logarithms

With the simplified equations in hand, where both sides of the logarithm equation share the same base, we can use another logarithm property to simplify further:

• If \[\log_b(M) = \log_b(N)\], then \[M = N\]

This means that once the equation has the same base on both sides, you can drop the logarithms and directly equate the arguments. In our case, the equation \[\log_3\left(\frac{x}{2}\right) = \log_3(77)\]leads us to:

• Equate \[\frac{x}{2} = 77\]
• This step directly simplifies logarithmic equations into a manageable algebraic form

This property allows us to transition from a logarithmic form into a straightforward algebraic equation.

Algebraic Manipulation to Solve for the Variable

After equating the arguments of both sides of the equation, we are left with a simple algebraic equation to solve. We use basic algebraic manipulation to find the value of \(x\):

• Start by isolating \(x\) in the equation \[\frac{x}{2} = 77\]
• Multiply both sides by 2 to solve for \(x\): \[x = 77 \times 2\]
• Calculate to find \[x = 154\]

This step involves using simple algebraic techniques like multiplication or division to isolate the variable, which in this case allows us to find that \(x = 154\). Algebraic manipulation is often the final step in solving logarithmic equations, providing the solution straightforwardly once the problem has been reduced through previous steps.