Sure! Let's solve the equation step-by-step to find an equivalent form. Given the logarithmic equation:
[tex]\[ \log(5x^3) - \log(x^2) = 2 \][/tex]
We can use the properties of logarithms to simplify the equation. Here is a detailed breakdown:
### Step 1: Apply the properties of logarithms
We know that [tex]\(\log a - \log b = \log \left(\frac{a}{b}\right)\)[/tex]. Using this property on the given equation:
[tex]\[ \log(5x^3) - \log(x^2) = \log \left(\frac{5x^3}{x^2}\right) \][/tex]
### Step 2: Simplify the argument inside the logarithm
Simplify the fraction inside the logarithm:
[tex]\[ \frac{5x^3}{x^2} = 5x^{3-2} = 5x \][/tex]
So, the equation now becomes:
[tex]\[ \log(5x) = 2 \][/tex]
### Step 3: Convert the logarithmic equation to its exponential form
Recall that [tex]\(\log_b a = c\)[/tex] is equivalent to [tex]\(b^c = a\)[/tex]. Here our base [tex]\(b\)[/tex] is 10 (common logarithm):
[tex]\[ 10^{\log(5x)} = 10^2 \][/tex]
### Conclusion
Given the equivalent forms listed in the question, the correct one is:
[tex]\[ 10^{\log \left(5x\right)} = 10^2 \][/tex]
Thus, the equivalent equation is
[tex]\[ 10^{\log 5 x^5}=10^2 \][/tex]
