Buszmenan

2022-10-11

Solve simultaneous logarithmic equations with different bases?

How do I solve these simultaneous equations?

$2lo{g}_{x}y+2lo{g}_{y}x=5$

$xy=8$

I've tried to convert the first formula to fraction form and continue from there, but I can't seem to get anywhere. I've tried to do

$x=8/y$

and substitute to the first equation, but I still can't seem to solve this. How do I go about in solving these types of equations?

How do I solve these simultaneous equations?

$2lo{g}_{x}y+2lo{g}_{y}x=5$

$xy=8$

I've tried to convert the first formula to fraction form and continue from there, but I can't seem to get anywhere. I've tried to do

$x=8/y$

and substitute to the first equation, but I still can't seem to solve this. How do I go about in solving these types of equations?

Samantha Braun

Beginner2022-10-12Added 9 answers

Say ${\mathrm{log}}_{x}y=a$

Therefore

$$a+\frac{1}{a}=\frac{5}{2}$$

$$2{a}^{2}-5a+2=0$$

$$(2a-1)(a-2)=0$$

$$a=2,\frac{1}{2}$$

Hence we have

$${\mathrm{log}}_{x}y=2,\frac{1}{2}$$

Now

$$xy=8$$

$$1+{\mathrm{log}}_{x}y={\mathrm{log}}_{x}8$$

Hence

$${\mathrm{log}}_{x}8-1=2$$

$${\mathrm{log}}_{x}8=3$$

$${\mathrm{log}}_{8}x=\frac{1}{3}$$

$$x={8}^{\frac{1}{3}}=2$$

Similarly we also have

$${\mathrm{log}}_{x}8-1=\frac{1}{2}$$

$${\mathrm{log}}_{x}8=\frac{3}{2}$$

$$x=4$$

The solutions hence follow.

Therefore

$$a+\frac{1}{a}=\frac{5}{2}$$

$$2{a}^{2}-5a+2=0$$

$$(2a-1)(a-2)=0$$

$$a=2,\frac{1}{2}$$

Hence we have

$${\mathrm{log}}_{x}y=2,\frac{1}{2}$$

Now

$$xy=8$$

$$1+{\mathrm{log}}_{x}y={\mathrm{log}}_{x}8$$

Hence

$${\mathrm{log}}_{x}8-1=2$$

$${\mathrm{log}}_{x}8=3$$

$${\mathrm{log}}_{8}x=\frac{1}{3}$$

$$x={8}^{\frac{1}{3}}=2$$

Similarly we also have

$${\mathrm{log}}_{x}8-1=\frac{1}{2}$$

$${\mathrm{log}}_{x}8=\frac{3}{2}$$

$$x=4$$

The solutions hence follow.

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

Which operation could we perform in order to find the number of milliseconds in a year??

$60\cdot 60\cdot 24\cdot 7\cdot 365$ $1000\cdot 60\cdot 60\cdot 24\cdot 365$ $24\cdot 60\cdot 100\cdot 7\cdot 52$ $1000\cdot 60\cdot 24\cdot 7\cdot 52?$ Tell about the meaning of Sxx and Sxy in simple linear regression,, especially the meaning of those formulas

Is the number 7356 divisible by 12? Also find the remainder.

A) No

B) 0

C) Yes

D) 6What is a positive integer?

Determine the value of k if the remainder is 3 given $({x}^{3}+k{x}^{2}+x+5)\xf7(x+2)$

Is $41$ a prime number?

What is the square root of $98$?

Is the sum of two prime numbers is always even?

149600000000 is equal to

A)$1.496\times {10}^{11}$

B)$1.496\times {10}^{10}$

C)$1.496\times {10}^{12}$

D)$1.496\times {10}^{8}$Find the value of$\mathrm{log}1$ to the base $3$ ?

What is the square root of 3 divided by 2 .

write $\sqrt[5]{{\left(7x\right)}^{4}}$ as an equivalent expression using a fractional exponent.

simplify $\sqrt{125n}$

What is the square root of $\frac{144}{169}$