How do you write as a single logarithm ${log}_{b} 5 + \frac{1}{3} {log}_{b} x$ $Log_b 5 +1/3 Log_bx$ ?

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How do you write as a single logarithm ${log}_{b} 5 + \frac{1}{3} {log}_{b} x$ $Log_b 5 +1/3 Log_bx$ ?

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Answer 1 · 2015-07-20T02:20:56

I found: ${log}_{b} (5 x^{\frac{1}{3}}) = {log}_{b} (5 \sqrt[3]{x})$ $log_b(5x^(1/3))=log_b(5root3(x))$

Explanation:

You can use the fact that:
$a log x = {log x}^{a}$ $alogx=logx^a$
and:
$log a + log b = log (a \times b)$ $loga+logb=log(axxb)$
So you get:
${log}_{b} (5) + {log}_{b} (x^{\frac{1}{3}}) =$ $log_b(5)+log_b(x^(1/3))=$
$= {log}_{b} (5 x^{\frac{1}{3}})$ $=log_b(5x^(1/3))$

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How do you write as a single logarithm ${log}_{b} 5 + \frac{1}{3} {log}_{b} x$ $Log_b 5 +1/3 Log_bx$ ?

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