To show that [tex]\( \log_{10} a - \log_{10} b = 2 \log_{10} c \)[/tex] implies [tex]\( a = b c^2 \)[/tex], we will proceed step-by-step using logarithmic properties.
1. Start with the given equation:
[tex]\[
\log_{10} a - \log_{10} b = 2 \log_{10} c
\][/tex]
2. Use the logarithm subtraction rule:
The property [tex]\(\log_{10} x - \log_{10} y = \log_{10} \left(\frac{x}{y}\right)\)[/tex] allows us to combine the left-hand side:
[tex]\[
\log_{10} \left(\frac{a}{b}\right) = 2 \log_{10} c
\][/tex]
3. Express the right-hand side:
The property [tex]\(k \log_{10} x = \log_{10} x^k\)[/tex] lets us rewrite the right-hand side:
[tex]\[
\log_{10} \left(\frac{a}{b}\right) = \log_{10} c^2
\][/tex]
4. Equate the arguments of the logarithms:
Since the expressions inside the [tex]\(\log_{10}\)[/tex] functions are equal, the arguments must be equal (logarithmic function is one-to-one):
[tex]\[
\frac{a}{b} = c^2
\][/tex]
5. Solve for [tex]\(a\)[/tex]:
Multiply both sides by [tex]\(b\)[/tex] to isolate [tex]\(a\)[/tex]:
[tex]\[
a = b c^2
\][/tex]
Thus, we have shown that given [tex]\(\log_{10} a - \log_{10} b = 2 \log_{10} c\)[/tex], it follows that [tex]\(a = b c^2\)[/tex].
