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?

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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?

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Answer 1 · 2016-11-21T03:15:48

$log (x) + log (y) = log (x y)$ $log(x)+log(y) = log(xy)$

Explanation:

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

$log (x) - log (y) = log (\frac{x}{y})$ $log(x)-log(y) = log(x/y)$
$log (x^{a}) = a log (x)$ $log(x^a) = alog(x)$
${log}_{x} (x) = 1$ $log_x(x) = 1$

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

Then ${log}_{a} (x y)$ $log_a(xy)$ is the unique value fulfilling $a^{{log}_{a} (x y)} = x y$ $a^(log_a(xy)) = xy$ .

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

$a^{{log}_{a} (x) + {log}_{a} (y)} = a^{{log}_{a} (x)} a^{{log}_{a} (y)} = x y$ $a^(log_a(x)+log_a(y)) = a^(log_a(x))a^(log_a(y))=xy$

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

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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?

1 Answer

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