Sure, let's solve the equation step by step.
The given equation is:
[tex]\[ \log_4(20x - 10) - 2 \log_4(3) = \log_4(4x - 10) \][/tex]
Step 1: Apply the Power Property of Logarithms
We know that [tex]\(a \log_b(c) = \log_b(c^a)\)[/tex]. Thus, we can rewrite [tex]\(2 \log_4(3)\)[/tex] as [tex]\(\log_4(3^2)\)[/tex]:
[tex]\[ \log_4(20x - 10) - \log_4(3^2) = \log_4(4x - 10) \][/tex]
Reason for this step: Power Property of Logarithms
Step 2: Apply the Quotient Property of Logarithms
Next, we use the property of logarithms that states [tex]\(\log_b(a) - \log_b(c) = \log_b(\frac{a}{c})\)[/tex]:
[tex]\[ \log_4\left(\frac{20x - 10}{9}\right) = \log_4(4x - 10) \][/tex]
Reason for this step: Quotient Property of Logarithms
Step 3: Set the arguments of the logarithms equal to each other
Since the logarithms with the same base are equal, their arguments must also be equal:
[tex]\[ \frac{20x - 10}{9} = 4x - 10 \][/tex]
Reason for this step: If [tex]\(\log_b(a) = \log_b(c)\)[/tex], then [tex]\(a = c\)[/tex]
Step 4: Solve the equation
Now we solve the equation [tex]\(\frac{20x - 10}{9} = 4x - 10\)[/tex]:
1. Multiply both sides by 9 to clear the fraction:
[tex]\[ 20x - 10 = 9(4x - 10) \][/tex]
2. Distribute the 9 on the right-hand side:
[tex]\[ 20x - 10 = 36x - 90 \][/tex]
3. Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 20x - 36x = -90 + 10 \][/tex]
[tex]\[ -16x = -80 \][/tex]
4. Divide both sides by -16:
[tex]\[ x = 5 \][/tex]
Therefore, the solution to the equation is:
[tex]\[ x = 5 \][/tex]
