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d) If
log_(10)y=(2)/(3)log_(10)x-2, express
y in terms of
x with no logarithmic expression.
2

d) If $\log _{10} y=\frac{2}{3} \log _{10} x-2$ , express $y$ in terms of $x$ with no logarithmic expression. $\newline$ $2$

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Quotient property of logarithms

Full solution

Q. d) If

\log _{10} y=\frac{2}{3} \log _{10} x-2

, express

y

in terms of

x

with no logarithmic expression.

\newline

2

Rewrite Logarithmic Equation: Rewrite the given logarithmic equation without the fraction: $\log_{10}y = \frac{2}{3}\log_{10}x - 2$ .
Use Power Property: Use the power property of logarithms to rewrite $(\frac{2}{3})\log_{10}x$ as $\log_{10}x^{(\frac{2}{3})}$ : $\log_{10}y = \log_{10}x^{(\frac{2}{3})} - 2$ .
Rewrite in Exponential Form: To remove the logarithms, rewrite the equation in exponential form: $10^{\log_{10}y} = 10^{\log_{10}x^{\frac{2}{3}} - 2}$ .
Simplify Left Side: Simplify the left side using the fact that $10^{\log_{10}y}$ is just $y$ : $y = 10^{\log_{10}x^{\frac{2}{3}} - 2}$ .
Split Right Side: Split the right side into two parts using the property of exponents: $y = \frac{10^{\log_{10}x^{\frac{2}{3}}}}{10^2}.$
Simplify First Part: Simplify the first part using the fact that $10^{\log_{10}A}$ is just $A$ : $y = \frac{x^{\frac{2}{3}}}{10^2}$ .
Final Expression for y: Finally, write the expression for y: $y = \frac{x^{\frac{2}{3}}}{100}.$

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