# Logarithm Equation Question 3

This is an interesting question which I came across under Additional Mathematics (A-Math):

Find the value of x.

$(log_x \sqrt{3})(log_x\sqrt{8})= \frac{3}{2}log_x\frac{1}{\sqrt{2}}$

Who is courageous to work on this question?

Update:

$(log_x \sqrt{3})(log_x\sqrt{8})= \frac{3}{2}log_x\frac{1}{\sqrt{2}}$

(Apply Power Law)

$(\frac{1}{2}log_x3)(\frac{1}{2}log_x8)= \frac{3}{2}log_x\frac{1}{\sqrt{2}}$

$\frac{1}{4}(log_x3)(log_x8)= \frac{3}{2}log_x\frac{1}{\sqrt{2}}$

(multiply by 4 on both sides of equal sign)

$(log_x3)(log_x8)= 6log_x\frac{1}{\sqrt{2}}$

(Apply Power Law)
$(log_x3)(log_x8)= log_x(\frac{1}{\sqrt{2}})^6$

$(log_x3)(log_x8)= log_x\frac{1}{8}$

(Apply Quotient Law)
$(log_x3)(log_x8)= log_x1-log_x8$
$(log_x3)(log_x8)+log_x8=0$

(Factorise)

$(log_x8)(log_x3+1)=0$

$(log_x8)=0 (NA)$$or log_x3+1=0$

$log_x3=-1,3=x^-^1,3=\frac {1}{x},3x=1,x= \frac {1}{3}$

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### Ai Ling Ong

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### 5 Responses to Logarithm Equation Question 3

1. (log-baseX-root3)(log-baseX-root8)=1.5(log-baseX-1/root2)

(0.5log-baseX-3)(1.5log-baseX-2)=1.5(log-baseX-2^-0.5)

[cancel 1.5 from both sides]

(0.5log-baseX-3)(log-baseX-2)= -0.5(log-baseX-2)

[cancel 0.5 from both sides]

(log-baseX-3)(log-baseX-2) + (log-baseX-2)= 0

(log-baseX-2) (log-baseX-3 + 1) = 0

therefore,

(log-baseX-2)=0 or (log-baseX-3 + 1)=0

X^0= 2 (rejected) (log-baseX-3)= -1
X^(-1) = 3
thus X= 1/3

hope i didnt make any mistakes
Logarithm rocks! XDD

2. awesome!
Tks for the effort.

Hie you are just an amazing lady thanks for everything.i will chat with you next time,tell me about your place the temperature & the environment

3. ahm97sic says:

This question is fun. Perhaps, you will like to let your students to try to solve the question.

Solve the equation

(2/x)^(log n 4) - (3/x)^(log n 9) = 0

Regards,

ahm97sic

4. ahm97sic says:

This is another interesting question. Perhaps, you will like to let your students to try to solve the question.

Solve (log a X)^(log b X) = X

where a, b are positive real numbers except 1, leave

the answer in terms of a and b.

Regards,

ahm97sic