Let's solve the given equation step-by-step.
Given equation:
[tex]$ \log(x^4) - \log(x^3) = \log(5x) - \log(2x) $[/tex]
### Step 1: Simplify both sides by using the properties of logarithms
We know that the difference of two logarithms can be expressed as the logarithm of their quotient. This gives us:
- Left-hand side (LHS):
[tex]$ \log(x^4) - \log(x^3) = \log\left(\frac{x^4}{x^3}\right) = \log(x) $[/tex]
- Right-hand side (RHS):
[tex]$ \log(5x) - \log(2x) = \log\left(\frac{5x}{2x}\right) = \log\left(\frac{5}{2}\right) $[/tex]
So the equation simplifies to:
[tex]$ \log(x) = \log\left(\frac{5}{2}\right) $[/tex]
### Step 2: Use the property of logarithms that states if [tex]\(\log(a) = \log(b)\)[/tex], then [tex]\(a = b\)[/tex]
From the simplified equation:
[tex]$ \log(x) = \log\left(\frac{5}{2}\right) $[/tex]
We can deduce:
[tex]$ x = \frac{5}{2} $[/tex]
### Step 3: Verify the solution
We should verify that our solution satisfies the original equation. Substituting [tex]\( x = \frac{5}{2} \)[/tex] back into the original equation:
- Compute LHS:
[tex]$ \log\left(\left(\frac{5}{2}\right)^4\right) - \log\left(\left(\frac{5}{2}\right)^3\right) $[/tex]
Simplifies to:
[tex]$ \log\left(\frac{625}{16}\right) - \log\left(\frac{125}{8}\right) = \log\left(\frac{625}{16} \cdot \frac{8}{125}\right) = \log\left(\frac{5}{2}\right) $[/tex]
- Compute RHS:
[tex]$ \log(5 \cdot \frac{5}{2}) - \log(2 \cdot \frac{5}{2}) $[/tex]
Simplifies to:
[tex]$ \log\left(\frac{25}{2}\right) - \log(5) = \log\left(\frac{25}{2} \cdot \frac{1}{5}\right) = \log\left(\frac{5}{2}\right) $[/tex]
Both sides are equal, confirming that our solution satisfies the original equation.
Thus, the solution to the equation is:
[tex]$ x = \frac{5}{2} $[/tex]
