# Thread: Logs - Without calculator

1. ## Logs - Without calculator

Hi all,

What's the best way to work out log's without a calculator / other aids?

The question is $\log_{8}\frac{1}{4}$ what is the answer.

I know the answer is $-\frac{2}{3}$ and I can work it out simply on a calculator. How do I do it without though?

I'm not sure the working method (without trial and error), any help would be appreciated. Sorry if it's pretty basic.

2. Originally Posted by Peleus
Hi all,

What's the best way to work out log's without a calculator / other aids?

The question is $\log_{8}\frac{1}{4}$ what is the answer.

I know the answer is $-\frac{2}{3}$ and I can work it out simply on a calculator. How do I do it without though?

I'm not sure the working method (without trial and error), any help would be appreciated. Sorry if it's pretty basic.
use the change of base formula

recall, $\log_a b = \frac {\log_c b}{\log_c a}$

so $\log_8 \frac 14 = \log_8 (4^{-1}) = - \log_84 = - \frac {\log_24}{\log_28} = - \frac 23$

for the second equal sign, i used the rule: $\log_a (x^n) = n \log_a x$

3. Awesome, makes it a lot clearer. Thank you.

Sometimes trying to self teach yourself things you run into basic questions you don't know the answer for

Appreciate it.

4. let $y = \log_8\left(\frac{1}{4}\right)$

change to an exponential equation ...

$8^y = \frac{1}{4}$

$(2^3)^y = 2^{-2}$

$2^{3y} = 2^{-2}$

$3y = -2$

$y = -\frac{2}{3}$

5. I'll put another one in here instead of starting up a new thread.

I need to simplify this logarithm. (By solving)

$\log_{2}18 - 2\log_{2}3$

So far I change that (I don't know if it breaks rules or not) to ...

$-2\log_{2}\frac{18}{3} = -2\log_{2}6$

That gives me a nasty decimal answer which I know isn't right. Correct answer is listed as one but I'm not sure how to manipulate it to cancel it out.

Any ideas?

Thank you again for your help so far.

6. Originally Posted by Peleus
I'll put another one in here instead of starting up a new thread.

I need to simplify this logarithm. (By solving)

$\log_{2}18 - 2\log_{2}3$

So far I change that (I don't know if it breaks rules or not) to ...

$-2\log_{2}\frac{18}{3} = -2\log_{2}6$

That gives me a nasty decimal answer which I know isn't right. Correct answer is listed as one but I'm not sure how to manipulate it to cancel it out.

Any ideas?

Thank you again for your help so far.
yes, you broke the rules. you have to change $2 \log_2 3$ to $\log_2 (3^2) = \log_2 9$ BEFORE combining the logs.

look at the rule again for simplifying the difference of two logarithms. note that the constant in front of each is 1

P.S. you should post new questions in a new thread

7. Excellent.

I know how to handle numbers in front now, thank you =)

I'll try and put new questions in new threads now, didn't want to clog the forums with my basic questions though

8. Originally Posted by Peleus
Excellent.

I know how to handle numbers in front now, thank you =)