Answer :
Certainly! Let's solve the equation \(14 = \log_5(2x - 3)\) by graphing, guided by the student’s given steps.
### Step-by-Step Solution:
#### Step 1:
Identify the two equations to be graphed:
- The horizontal line representing the constant value: \(y_1 = 14\)
- The logarithmic function: \(y_2 = \log_5(2x - 3)\)
#### Step 2:
Use the change of base formula to rewrite the equations:
- For the horizontal line, it remains: \(y_1 = 14\)
- For the logarithmic function, we apply the change of base formula:
[tex]\[ \log_5(2x - 3) = \frac{\log(2x - 3)}{\log 5} \][/tex]
So, the equation transforms into: \(y_2 = \frac{\log(2x - 3)}{\log 5}\)
#### Step 3:
Graph the two equations on the same coordinate plane:
- The equation \(y_1 = 14\) is a horizontal line at \(y = 14\).
- The equation \(y_2 = \frac{\log(2x - 3)}{\log 5}\) is a transformed logarithmic function.
#### Step 4:
Find the intersection of the graphs:
- The solution to the equation \(14 = \log_5(2x - 3)\) is where the graphs of \(y_1\) and \(y_2\) intersect.
- This would be the \(x\)-coordinate of the point where the line \(y_1 = 14\) intersects the curve \(y_2 = \frac{\log(2x - 3)}{\log 5}\).
### Detailed clarification:
To find the point of intersection:
1. Set the equations equal to each other:
[tex]\[ 14 = \frac{\log(2x - 3)}{\log 5} \][/tex]
2. Solve for \(2x - 3\):
Multiply both sides by \(\log 5\):
[tex]\[ 14 \log 5 = \log(2x - 3) \][/tex]
3. Rewrite the equation in exponential form to solve for \(2x - 3\):
[tex]\[ 10^{14 \log 5} = 2x - 3 \][/tex]
4. Simplify the exponential term using properties of logarithms:
[tex]\[ 10^{14 \log 5} = (10^{\log 5})^{14} = 5^{14} \][/tex]
Thus, we have:
[tex]\[ 5^{14} = 2x - 3 \][/tex]
5. Solve for \(x\):
[tex]\[ 2x = 5^{14} + 3 \][/tex]
[tex]\[ x = \frac{5^{14} + 3}{2} \][/tex]
Therefore, the [tex]\(x\)[/tex] value where the two graphs intersect and which satisfies the original equation [tex]\(14 = \log_5(2x - 3)\)[/tex] is [tex]\(x = \frac{5^{14} + 3}{2}\)[/tex].
### Step-by-Step Solution:
#### Step 1:
Identify the two equations to be graphed:
- The horizontal line representing the constant value: \(y_1 = 14\)
- The logarithmic function: \(y_2 = \log_5(2x - 3)\)
#### Step 2:
Use the change of base formula to rewrite the equations:
- For the horizontal line, it remains: \(y_1 = 14\)
- For the logarithmic function, we apply the change of base formula:
[tex]\[ \log_5(2x - 3) = \frac{\log(2x - 3)}{\log 5} \][/tex]
So, the equation transforms into: \(y_2 = \frac{\log(2x - 3)}{\log 5}\)
#### Step 3:
Graph the two equations on the same coordinate plane:
- The equation \(y_1 = 14\) is a horizontal line at \(y = 14\).
- The equation \(y_2 = \frac{\log(2x - 3)}{\log 5}\) is a transformed logarithmic function.
#### Step 4:
Find the intersection of the graphs:
- The solution to the equation \(14 = \log_5(2x - 3)\) is where the graphs of \(y_1\) and \(y_2\) intersect.
- This would be the \(x\)-coordinate of the point where the line \(y_1 = 14\) intersects the curve \(y_2 = \frac{\log(2x - 3)}{\log 5}\).
### Detailed clarification:
To find the point of intersection:
1. Set the equations equal to each other:
[tex]\[ 14 = \frac{\log(2x - 3)}{\log 5} \][/tex]
2. Solve for \(2x - 3\):
Multiply both sides by \(\log 5\):
[tex]\[ 14 \log 5 = \log(2x - 3) \][/tex]
3. Rewrite the equation in exponential form to solve for \(2x - 3\):
[tex]\[ 10^{14 \log 5} = 2x - 3 \][/tex]
4. Simplify the exponential term using properties of logarithms:
[tex]\[ 10^{14 \log 5} = (10^{\log 5})^{14} = 5^{14} \][/tex]
Thus, we have:
[tex]\[ 5^{14} = 2x - 3 \][/tex]
5. Solve for \(x\):
[tex]\[ 2x = 5^{14} + 3 \][/tex]
[tex]\[ x = \frac{5^{14} + 3}{2} \][/tex]
Therefore, the [tex]\(x\)[/tex] value where the two graphs intersect and which satisfies the original equation [tex]\(14 = \log_5(2x - 3)\)[/tex] is [tex]\(x = \frac{5^{14} + 3}{2}\)[/tex].
