Problem 128
Question
Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set Verify this value by direct substitution into the equation. $$ \log _{3}(3 x-2)=2 $$
Step-by-Step Solution
Verified Answer
The solution set of the equation is \(x=\frac{11}{3}\) or approximately \(3.67\). This is confirmed by graphing and substitution.
1Step 1: Understanding the log equation
The given equation is \( \log_{3}(3x-2)=2 \). This states that the base 3 raised to the power 2 equals to \(3x-2\). The equivalent exponential form of the log equation can be written as \(3^2=3x-2\)
2Step 2: Solving for x
Let's solve the exponential equation \(3^2=3x-2\). Simplify this to get \(9=3x-2\). Solving for x yields, \(x=\frac{9+2}{3}= \frac{11}{3}\) or approximately \(3.67\)
3Step 3: Graphing and verifying the solution
Plot \(y=\log_{3}(3x-2)\) and \(y=2\) using a graphing tool. The x-coordinate of the point of intersection is the solution for x. Validate this value by substituting \(x=\frac{11}{3}\) into the original log equation, the left-hand side should equal the right-hand side.
Key Concepts
Logarithmic FunctionsIntersection PointsExponential Equations
Logarithmic Functions
Logarithmic functions are an important concept in mathematics and can be best understood by considering their relationship with exponential functions. A logarithmic function, expressed as \( \log_b(a) \), represents the power to which a base \( b \) must be raised to produce the number \( a \). For example, \( \log_3(9) = 2 \) because \( 3^2 = 9 \).
You can think of logarithms as the inverse of exponentials. This relationship is crucial when converting logarithmic equations to exponential form and vice versa. This is vital for solving logarithmic equations because it allows us to rewrite and understand them in a form that might be more familiar or easier to manipulate.
You might encounter logarithms across various applications like scientific calculations, computer science, and even in music theory, any context where scaling or exponential growth needs to be understood or manipulated.
You can think of logarithms as the inverse of exponentials. This relationship is crucial when converting logarithmic equations to exponential form and vice versa. This is vital for solving logarithmic equations because it allows us to rewrite and understand them in a form that might be more familiar or easier to manipulate.
You might encounter logarithms across various applications like scientific calculations, computer science, and even in music theory, any context where scaling or exponential growth needs to be understood or manipulated.
Intersection Points
When graphing equations, finding intersection points is a useful method to determine solutions graphically. An intersection point occurs where the graphs of two equations meet, meaning they have the same coordinate values at that point.
In the context of the given equation \( \log_{3}(3x-2) = 2 \), you plot two graphs: one for \( y = \log_3(3x-2) \) and another for \( y = 2 \). The x-coordinate of their intersection point provides the solution \( x \) to the equation because it simultaneously satisfies both equations.
Some helpful points to remember when finding intersection points:
In the context of the given equation \( \log_{3}(3x-2) = 2 \), you plot two graphs: one for \( y = \log_3(3x-2) \) and another for \( y = 2 \). The x-coordinate of their intersection point provides the solution \( x \) to the equation because it simultaneously satisfies both equations.
Some helpful points to remember when finding intersection points:
- Ensure both functions are plotted accurately within the same viewing window.
- Look for where the graphs cross each other; this represents shared solutions.
- Use graphing tools to improve precision, especially where the intersection might not be apparent at first glance.
Exponential Equations
Exponential equations involve variables in exponents and are expressed in the form \( b^x = y \). Converting logarithmic equations into exponential form, like transforming \( \log_{3}(3x-2) = 2 \) into \( 3^2 = 3x - 2 \), helps in finding their solutions.
The key step in solving exponential equations is to make them easy to work with by aligning the base with the other side of the equation. For example, in \( 3^2 = 3x - 2 \), the base 3 is crucial because it helps eradicate the variable's exponent through simplification.
In our solution, exponentiating helps us isolate \( x \) effectively, leading to the solution \( x = \frac{11}{3} \). Always ensure upon solving, you validate by substituting back into the original equation, ensuring both sides equal, confirming the solution is correct.
The key step in solving exponential equations is to make them easy to work with by aligning the base with the other side of the equation. For example, in \( 3^2 = 3x - 2 \), the base 3 is crucial because it helps eradicate the variable's exponent through simplification.
In our solution, exponentiating helps us isolate \( x \) effectively, leading to the solution \( x = \frac{11}{3} \). Always ensure upon solving, you validate by substituting back into the original equation, ensuring both sides equal, confirming the solution is correct.
Other exercises in this chapter
Problem 127
Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the
View solution Problem 128
Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$ f(x)=\ln x, g(x)=\ln (x+3) $$
View solution Problem 129
Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$ f(x)=\ln x, g(x)=\ln x+3 $$
View solution Problem 129
Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the
View solution