Certainly! Let's solve the given logarithmic equation step by step to find the equivalent equation.
We start with the equation:
[tex]\[
\log(5x^3) - \log(x^2) = 2
\][/tex]
Step 1: Use the properties of logarithms to simplify the expression on the left-hand side. One of the properties states that [tex]\(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\)[/tex]. Therefore, we can write:
[tex]\[
\log(5x^3) - \log(x^2) = \log\left(\frac{5x^3}{x^2}\right)
\][/tex]
Step 2: Simplify the argument of the logarithm. By dividing [tex]\(5x^3\)[/tex] by [tex]\(x^2\)[/tex], we get:
[tex]\[
\frac{5x^3}{x^2} = 5x^{3-2} = 5x
\][/tex]
So the equation becomes:
[tex]\[
\log(5x) = 2
\][/tex]
Step 3: Rewrite the equation in its exponential form. Recall that if [tex]\(\log_b(y) = x\)[/tex], then [tex]\(y = b^x\)[/tex]. Here, [tex]\(b\)[/tex] (the base of the logarithm) is understood to be 10, because it's the common logarithm. Therefore:
[tex]\[
5x = 10^2
\][/tex]
Step 4: Write the equivalent exponential equation involving the logarithm. We correctly expressed [tex]\(5x\)[/tex] in terms of the given comparison with base 10 logarithm:
[tex]\[
10^{\log(5x)} = 10^2
\][/tex]
Using the properties of equations involving logarithms and exponentials, we derive:
[tex]\[
10^{\log\left(\frac{5x^3}{x^2}\right)} = 10^2
\][/tex]
Thus, the correct equivalent equation is:
[tex]\[
10^{\log\left(\frac{5x^3}{x^2}\right)} = 10^2
\][/tex]
So, the answer is:
[tex]\[
10^{\log \frac{5 x^3}{x^2}}=10^2
\][/tex]
