Find logb[sqrt10b] given logb2=.3562 and logb5=0.8271

[Bladesofhalo]

The problem states "Find logb[sqrt10b] given logb2=.3562 and logb5=0.8271.

So after rewriting, it would be:
b.3562=2
b.8271=5

Am I proceeding the right way? Do I have to solve using natural logarithms? Or am I looking at this problem completely the wrong way? I dont need the answer outright, just need a hint if possible.

 

[Deleted member 4993]

Re: Logarithm problem

The problem states "Find logb[sqrt10b] given logb2=.3562 and logb5=0.8271.

So after rewriting, it would be:
b.3562=2
b.8271=5

Am I proceeding the right way? Do I have to solve using natural logarithms? Or am I looking at this problem completely the wrong way? I dont need the answer outright, just need a hint if possible.

Is your problem:

\(\displaystyle Find \,\ Log_b\sqrt{10b}\,\ given \,\ Log_b2 \,\ = \,\ 0.3562\,\ and \,\ Log_b5 = 0.8271\) ?

 

[Bladesofhalo]

Re: Logarithm problem

Yes, sorry about not writing it more clearly.

And it should also read like this:
b[sup:3jwql0sn].3562[/sup:3jwql0sn]=2
b[sup:3jwql0sn].8271[/sup:3jwql0sn]=5

 

[soroban]

Re: Logarithm problem

Hello, Bladesofhalo!

This is an antiquated problem . . .

\(\displaystyle \text{Find }\log_b(\sqrt{10b}),\text{ given: }\,\log_b(2)\:=\:0.3562\,\text{ and }\,\log_b(5)\:=\:0.8271\)

\(\displaystyle \text{We have: }\;\log_b(\sqrt{10b})\)

. . \(\displaystyle =\;\log_b(10b)^{\frac{1}{2}}\)

. . \(\displaystyle =\; \frac{1}{2}\log_b(10b)\)

. . \(\displaystyle = \;\frac{1}{2}\bigg[\log_b(10) + \log_b(b)\bigg]\)

. . \(\displaystyle =\;\frac{1}{2}\bigg[\log(2\cdot5) + 1\bigg]\)

. . \(\displaystyle = \;\frac{1}{2}\bigg[\log_b(2) + \log_b(5) + 1\bigg]\)

. . \(\displaystyle = \;\frac{1}{2}\bigg[0.3562 + 0.8271 + 1\bigg]\)

. . \(\displaystyle = \;\frac{1}{2}\bigg[2.1833\bigg]\)

. . \(\displaystyle =\;1.09165\)