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

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

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Product property of logarithms

Full solution

Q. If

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

, express

y

in terms of

x

with no logarithmic expression.

Rewrite Equation with Power Property: Rewrite the given equation using the power property of logarithms. $\newline$ $\log_{10}y = \frac{2}{3}\log_{10}x - 2$
Convert to Exponential Form: Convert the logarithmic equation to its exponential form. $\newline$ $10^{\log_{10}y} = 10^{\left(\frac{2}{3}\right)\log_{10}x - 2}$
Simplify Left Side: Simplify the left side of the equation using the fact that $10^{\log_{10}y} = y$ . $\newline$ $y = 10^{\left(\frac{2}{3}\log_{10}x - 2\right)}$
Split Right Side: Split the right side of the equation into two parts using the property of exponents $a^{(m - n)} = \frac{a^m}{a^n}$ . $\newline$ $y = \frac{10^{(\frac{2}{3}\log_{10}x)}}{10^2}$
Simplify Further: Simplify $10^2$ to $100$ . $\newline$ $y = \frac{10^{(\frac{2}{3}\log_{10}x)}}{100}$
Convert Back to x: Convert $10^{(\frac{2}{3})\log_{10}x}$ back to x using the power property of logarithms $(a^{\log_a(b)} = b)$ . $\newline$ $y = \frac{x^{\frac{2}{3}}}{100}$

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