Logarithm question involving different base

Question

Calculate the values of $z$ for which $\log_3 z = 4\log_z3$.

To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help you if you show that you've tried the problem yourself. Also, many would consider your post rude because it is a command ("Calculate"), not a request for help, so please consider rewriting it. — Zev Chonoles, May 20 '13 at 11:47

score 2 · Answer 1 · answered May 20 '13 at 11:46

2

Hint : use the formula for the change of base $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$ to change the base $z$ of the expression on the right.

answered May 20 '13 at 11:46

sigmatau

2,622

what is the value for k ? – Pranoy Bhatnagar May 20 '13 at 11:50
it is arbitrary. In this case just choose it to be $10$ so the equation is equivalent to $\log_b(x) = \frac{\log(x)}{\log(b)}$ , since the $\log$ function is in Base $10$ by definition. – sigmatau May 20 '13 at 11:53

score 0 · Answer 2 · edited May 20 '13 at 12:01

0

hint :

$$\frac{\log (z)}{\log (3)}=4\frac{\log (3)}{\log (z)}$$

$$(\log (z))^2=4(\log (3))^2$$

edited May 20 '13 at 12:01

Zev Chonoles

129,973

answered May 20 '13 at 11:48

mnsh

5,875

1

Please use \log. Writing log is interpreted as a product of variables named "l", "o", and "g", just as it is when you write ab in math mode, for example. The name of the function should be upright and have proper spacing, and that is what \log does. There are similar commands \sin, \cos, \exp, etc. – Zev Chonoles May 20 '13 at 12:03
1

@ZevChonoles:sorry about the wrong next time iam going to write correctly – mnsh May 20 '13 at 12:24

score 0 · Accepted Answer · answered May 20 '13 at 11:49

Hint

$$\log_3 z = 4\log_z3 \Rightarrow \dfrac{\log z}{\log 3}=4\dfrac{\log 3}{\log z}$$

Answer (don't look until you try the hint) $$\color{lightgrey}{\dfrac{\log z}{\log 3}=4\dfrac{\log 3}{\log z}\Rightarrow (\log z )^2=4(\log 3)^2 \Rightarrow \log z =2\log 3 \Rightarrow \log z =\log 3^2 \Rightarrow z = 3^2}$$

$\color{lightgrey}{\text{Therefore} \ z=9.}$

score 0 · Answer 4 · edited May 20 '13 at 12:00

0

$\frac{\log z}{\log 3}=4\frac{\log 3}{\log z}$

$(\log z)^2=(\log 3)(4\log 3)$

$\log z=\sqrt{(\log3)(4\log 3)}$

$ z=10 \exp\left(\sqrt{4(\log 3)^2}\right)$

edited May 20 '13 at 12:00

Zev Chonoles

129,973

answered May 20 '13 at 11:56

Teboho Tsotetsi

9

1

Please use \log. Writing log is interpreted as a product of variables named "l", "o", and "g", just as it is when you write ab in math mode, for example. The name of the function should be upright and have proper spacing, and that is what \log does. There are similar commands \sin, \cos, \exp, etc. – Zev Chonoles May 20 '13 at 12:03

Logarithm question involving different base

4 Answers4