How do you simplify ${log}_{2} (\frac{x^{6} y^{7}}{8})$ $\log _ { 2} ( \frac { x ^ { 6} y ^ { 7} } { 8} )$ ?

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How do you simplify ${log}_{2} (\frac{x^{6} y^{7}}{8})$ $\log _ { 2} ( \frac { x ^ { 6} y ^ { 7} } { 8} )$ ?

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Answer 1 · 2017-06-15T02:47:20

The expression equals $6 {log}^{2} x + 7 {log}^{2} y - 3$ $6log^2x + 7log^2y - 3$

Explanation:

Use the following laws of logarithms to simplify the above problem

$∙ {log}_{a} (n m) = {log}_{a} n + {log}_{a} m$ $• log_a(nm) = log_a n + log_a m$
$∙ {log}_{a} (\frac{n}{m}) = {log}_{a} n - {log}_{a} m$ $•log_a(n/m) = log_an - log_a m$
$∙ log (a^{n}) = n log a$ $•log(a^n) = nloga$

Now apply them:

$= {log}_{2} (x^{6} y^{7}) - {log}_{2} 8$ $=log_2(x^6y^7) - log_2 8$

$= {log}_{2} x^{6} + {log}_{2} y^{7} - 3$ $= log_2 x^6 + log_2 y^7 - 3$

$= 6 {log}_{2} x + 7 {log}_{2} y - 3$ $=6log_2x + 7log_2y - 3$

Hopefully this helps!

Answer 2 · 2017-06-15T02:54:39

${log}_{2} (x^{6}) + {log}_{2} (y^{7}) - 3$ $log_2(x^6)+log_2(y^7)-3$

Explanation:

Use log properties:
Product Rule:
$log x + log y = log x y$ $logx+logy=logxy$

Quotient Rule:
$log x - log y = log (\frac{x}{y})$ $logx-logy=log(x/y)$

Split the fraction into two separate logs using the Quotient Rule.
${log}_{2} (\frac{x^{6} y^{7}}{8})$ $log_2((x^6y^7)/8)$
${log}_{2} (x^{6} y^{7}) - {log}_{2} (8)$ $log_2(x^6y^7)-log_2(8)$

Split $(x^{6} y^{7})$ $(x^6y^7)$ into two separate logs using the Product Rule.
${log}_{2} (x^{6} y^{7}) - {log}_{2} (8)$ $log_2(x^6y^7)-log_2(8)$
${log}_{2} (x^{6}) + {log}_{2} (y^{7}) - {log}_{2} (8)$ $log_2(x^6)+log_2(y^7)-log_2(8)$

Simplify $- {log}_{2} (8)$ $-log_2(8)$ .
${log}_{2} (x^{6}) + {log}_{2} (y^{7}) - {log}_{2} (8)$ $log_2(x^6)+log_2(y^7)-log_2(8)$
${log}_{2} (x^{6}) + {log}_{2} (y^{7}) - 3$ $log_2(x^6)+log_2(y^7)-3$

${log}_{2} (x^{6}) + {log}_{2} (y^{7}) - 3$ $log_2(x^6)+log_2(y^7)-3$ is your simplified expression.

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How do you simplify ${log}_{2} (\frac{x^{6} y^{7}}{8})$ $\log _ { 2} ( \frac { x ^ { 6} y ^ { 7} } { 8} )$ ?

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How do you simplify ${log}_{2} (\frac{x^{6} y^{7}}{8})$ $\log _ { 2} ( \frac { x ^ { 6} y ^ { 7} } { 8} )$ ?