Questions: You are only allowed to use integer exponents. - Use sqrt(...) to deal with fractional exponents. - Make sure your final answer is written as a single logarithm. a. 7 log (x)+5 log (z)-1/2 log (y)=log ((x^7 z^5)/sqrt(y)) b. 1/2 log (x)-5 log (y)-7 log (z)=log (sqrt(x)/(y^5 z^7)) c. 5/2 log (x)-7/2 log (y)-5/2 log (z)=log (sqrt(x^5)/(y sqrt(7/2) sqrt(z^5))) Hint - Do you have a sum of logarithms? Use the fact that log (a)+log (b)=log (a b). - Do you have a difference of logarithms? Use the fact that log (a)-log (b)=log (a/b). - Remember that a^(m/n)=sqrt[n](a^m).

Solution Steps

To solve these logarithmic expressions, we will use the properties of logarithms to combine them into a single logarithm. Specifically, we will use the properties:

1. \(\log(a) + \log(b) = \log(ab)\)
2. \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\)
3. \(k \log(a) = \log(a^k)\)

We start with the expression \(7 \log(x) + 5 \log(z) - \frac{1}{2} \log(y)\). Using the properties of logarithms, we can combine these terms into a single logarithm: \[7 \log(x) + 5 \log(z) - \frac{1}{2} \log(y) = \log(x^7) + \log(z^5) - \log(y^{1/2})\] \[= \log(x^7 z^5) - \log(y^{1/2})\] \[= \log\left(\frac{x^7 z^5}{y^{1/2}}\right)\] \[= \log\left(\frac{x^7 z^5}{\sqrt{y}}\right)\]

We start with the expression \(\frac{1}{2} \log(x) - 5 \log(y) - 7 \log(z)\). Using the properties of logarithms, we can combine these terms into a single logarithm: \[\frac{1}{2} \log(x) - 5 \log(y) - 7 \log(z) = \log(x^{1/2}) - \log(y^5) - \log(z^7)\] \[= \log\left(\frac{x^{1/2}}{y^5}\right) - \log(z^7)\] \[= \log\left(\frac{x^{1/2}}{y^5 z^7}\right)\] \[= \log\left(\frac{\sqrt{x}}{y^5 z^7}\right)\]

We start with the expression \(\frac{5}{2} \log(x) - \frac{7}{2} \log(y) - \frac{5}{2} \log(z)\). Using the properties of logarithms, we can combine these terms into a single logarithm: \[\frac{5}{2} \log(x) - \frac{7}{2} \log(y) - \frac{5}{2} \log(z) = \log(x^{5/2}) - \log(y^{7/2}) - \log(z^{5/2})\] \[= \log\left(\frac{x^{5/2}}{y^{7/2}}\right) - \log(z^{5/2})\] \[= \log\left(\frac{x^{5/2}}{y^{7/2} z^{5/2}}\right)\] \[= \log\left(\frac{\sqrt{x^5}}{y^{7/2} \sqrt{z^5}}\right)\]

We start with the expression: \[ 7 \log(x) + 5 \log(z) - \frac{1}{2} \log(y) \] Using the properties of logarithms, we combine the terms: \[ = \log(x^7) + \log(z^5) - \log(y^{1/2}) = \log\left(\frac{x^7 z^5}{\sqrt{y}}\right) \] Substituting \(x = 2\), \(y = 3\), and \(z = 4\): \[ = \log\left(\frac{2^7 \cdot 4^5}{\sqrt{3}}\right) \approx 4.8789 \]

We start with the expression: \[ \frac{1}{2} \log(x) - 5 \log(y) - 7 \log(z) \] Using the properties of logarithms, we combine the terms: \[ = \log(x^{1/2}) - \log(y^5) - \log(z^7) = \log\left(\frac{x^{1/2}}{y^5 z^7}\right) \] Substituting \(x = 2\), \(y = 3\), and \(z = 4\): \[ = \log\left(\frac{\sqrt{2}}{3^5 \cdot 4^7}\right) \approx -6.4495 \]

We start with the expression: \[ \frac{5}{2} \log(x) - \frac{7}{2} \log(y) - \frac{5}{2} \log(z) \] Using the properties of logarithms, we combine the terms: \[ = \log(x^{5/2}) - \log(y^{7/2}) - \log(z^{5/2}) = \log\left(\frac{x^{5/2}}{y^{7/2} z^{5/2}}\right) \] Substituting \(x = 2\), \(y = 3\), and \(z = 4\): \[ = \log\left(\frac{\sqrt{x^5}}{y^{7/2} \sqrt{z^5}}\right) \approx -2.4225 \]

Final Answer