Question
Given $\log _{10} \mathrm{x}=2 \mathrm{a}$ and $\log _{10} \mathrm{y}=\frac{b}{2}$. If $\log _{10}^p=3 a-2 b$, express $\mathrm{P}$ in terms of $x$ and $y$.
✓
Answer
We know $10^a=x^{1 / 2}$
$10^{\mathrm{b} / 2}=\mathrm{y}$
$ \Rightarrow 10^{\mathrm{b}}=\mathrm{y}^2$
$\log _{10}^p=3 a-2 b$
$ \Rightarrow p=10^{3 a}-2 b$
$ \Rightarrow p=\left(10^3\right)^a \div\left(10^2\right)^b$
$ \Rightarrow p=(10 a)^3 \div\left(10^b\right)^2$
Substituting $10^{\mathrm{a}} \ 10^{\mathrm{b}}$, We get
$\Rightarrow \mathrm{p}=\left(\mathrm{x}^{1 / 2}\right)^3 \div\left(\mathrm{y}^2\right)^2$
$\Rightarrow \mathrm{p}=x^{\frac{3}{2}} \div y^4$
$\Rightarrow \mathrm{p}=\frac{x^{\frac{3}{2}}}{y^4}$
$10^{\mathrm{b} / 2}=\mathrm{y}$
$ \Rightarrow 10^{\mathrm{b}}=\mathrm{y}^2$
$\log _{10}^p=3 a-2 b$
$ \Rightarrow p=10^{3 a}-2 b$
$ \Rightarrow p=\left(10^3\right)^a \div\left(10^2\right)^b$
$ \Rightarrow p=(10 a)^3 \div\left(10^b\right)^2$
Substituting $10^{\mathrm{a}} \ 10^{\mathrm{b}}$, We get
$\Rightarrow \mathrm{p}=\left(\mathrm{x}^{1 / 2}\right)^3 \div\left(\mathrm{y}^2\right)^2$
$\Rightarrow \mathrm{p}=x^{\frac{3}{2}} \div y^4$
$\Rightarrow \mathrm{p}=\frac{x^{\frac{3}{2}}}{y^4}$
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