# Assuming x and y and z are positive ,use properties of logarithm to write expression as a sum or difference of logarithms or multiples of logarithms. 13.log x + log y 14. log x + log y?

Nov 21, 2016

$\log \left(x\right) + \log \left(y\right) = \log \left(x y\right)$

#### Explanation:

$\log \left(x\right) + \log \left(y\right) = \log \left(x + y\right)$ is a basic property of logarithms, along with

• $\log \left(x\right) - \log \left(y\right) = \log \left(\frac{x}{y}\right)$
• $\log \left({x}^{a}\right) = a \log \left(x\right)$
• ${\log}_{x} \left(x\right) = 1$

To derive the property used in the given question, recall that ${\log}_{a} \left(x\right)$ is defined as the unique value fulfilling ${a}^{{\log}_{a} \left(x\right)} = x$.

Then ${\log}_{a} \left(x y\right)$ is the unique value fulfilling ${a}^{{\log}_{a} \left(x y\right)} = x y$.

Now, taking $a$ to the power of ${\log}_{a} \left(x\right) + {\log}_{a} \left(y\right)$, we have

${a}^{{\log}_{a} \left(x\right) + {\log}_{a} \left(y\right)} = {a}^{{\log}_{a} \left(x\right)} {a}^{{\log}_{a} \left(y\right)} = x y$

As ${\log}_{a} \left(x\right) + {\log}_{a} \left(y\right)$ shares a property which is unique to ${\log}_{a} \left(x y\right)$, it must be that ${\log}_{a} \left(x\right) + {\log}_{a} \left(y\right) = {\log}_{a} \left(x y\right)$.