Solve for K given in this equation

9514 1404 393

Answer:

  logk(15) ≈ 0.902

  k ≈ 57.6  or  12.8  or  20.1  depending on how you calculate it

Step-by-step explanation:

The relevant rule of logarithms is ...

  [tex]\log_k(ab)=\log_k(a)+\log_k(b)[/tex]

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Using this rule, we have ...

  [tex]\log_k(15)=\log_k(3\cdot5)=\log_k(3)+\log_k(5)\\\\\log_k(15)=0.271+0.631\\\\ \boxed{\log_k(15)=0.902}[/tex]

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You will get different (inconsistent) results when you solve for k.

The applicable rule of logarithms is ...

  [tex]\log_k(a)=\dfrac{\log(a)}{\log(k)}[/tex]

Then we can find log(k) and take the antilog to get ...

  [tex]\log(k)=\dfrac{\log(a)}{\log_k(a)}\\\\k=(10^{\log(3)})^{1/\log_k(3)}=3^{1/0.271}\approx57.6\qquad\text{for $a=3$}\\\\ k=(10^{\log(5)})^{1/\log_k(5)}=5^{1/0.631}\approx12.8\qquad\text{for $a=5$}\\\\ k=(10^{\log(15)})^{1/\log_k(15)}=15^{1/0.902}\approx20.1\qquad\text{for $a=15$}[/tex]

Here, we have used base-10 logarithms, but the same result is obtained for any base. The different results simply serve to show that the numbers in the problem are not self-consistent.

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