Question for math; 2m3n-4/p-6

If logb2=x and logb3=y, evaluate the following in terms of x and y:

Alja [10]

$log_b{162} = x + 4y\\\\log_b324 = 2x+4y\\\\log_b\frac{8}{9} = 3x-2y\\\\\frac{log_b27}{log_b16} = 3y-4x$

Solution:

Given that,

$log_b2 = x\\\\log_b3 = y --------(i)$

Use the following log rules

Rule 1: $log_b(ac) = log_ba + log_bc$

Rule 2: $log_b\frac{a}{c} = log_ba - log_bc$

Rule 3: $log_ba^c = clog_ba$

$a) log_b{162}$

Break 162 down to primes:

$162 = 2^1 \times 3^4$

$log_b{162} =log_b 2^1. 3^4\\\\By\ rule\ 1\\\\ log_b{162} = log_b 2^1 +log_b 3^4\\\\By\ rule\ 3\\\\1log_b2 + 4log_b3\\\\1x+4y\\\\x+4y$

Thus we get,

$log_b162 = x + 4y$

Break 324 down to primes:

$324 = 2^2 \times 3^4$

$log_b324 = log_b 2^2.3^4\\\\By\ rule\ 1\\\\log_b324 = log_b2^2 + log_b3^4\\\\By\ rule\ 3\\\\log_b324 = 2log_b2 + 4log_b3\\\\From\ (i)\\\\log_b324 = 2x + 4y$

By rule 2

$log_b\frac{8}{9} = log_b8 - log_b9\\\\log_b\frac{8}{9} = log_b 2^3 - log_b3^2\\\\By\ rule\ 3\\\\log_b\frac{8}{9} = 3 log_b2 - 2log_b3\\\\From\ (i)\\\\log_b\frac{8}{9} = 3x - 2y$

By rule 2

$\frac{log_b27}{log_b16} = log_b27 - log_b16\\\\ \frac{log_b27}{log_b16} = log_b3^3 - log_b2^4\\\\By\ rule\ 2\\\\ \frac{log_b27}{log_b16} = 3log_b3 - 4log_b2 \\\\From\ (i)\\\\\frac{log_b27}{log_b16} = 3y - 4x$

Thus the given are evaluated in terms of x and y

3 0

2 years ago

PLEASE HELP!!!!! I don’t get it all :((((

asambeis [7]

The answer is 35 hope this helps

4 0

2 years ago

I will write about 18

Likurg_2 [28]

Answer

Painic-stricken, handle business, not a joke, yeah

Manners missing, travel different, no control, yeah

Time to listen, time to zip it, keep it closed

My description, highly gifted, take some notes, yeah

Lack of interest, why'd you visit? Hit the road, yeah

I'm kinda twisted, so keep your distance, be a ghost

Yeah, see I'm inventive, but quite the menace, you ain't know?

Well then I'm offended, let's jog your memories, here we go, yeah

I went from nobody to kinda famous

Hide my plaques inside the closet, I just can't explain it

My wife, she tells me that she's proud and thinks that I should hang 'em

But I just leave 'em on the ground right next to my self-hatred

Yeah, yeah, mental health, where's my mental health?

Diagnosed with OCD, what does that mean? Well, gather 'round

That means I obsessively obsess on things I think about

That means I might take a normal thought and think it's so profound (leave me alone)

Ruminating, fill balloons up full of doubt

3 0

3 years ago

Read 2 more answers

The function y=−0.25x+0.55 represents the percent y (in decimal form) of battery power remaining x hours after you turn on a lap

AURORKA [14]

Answer: a is 2.2, b is 0.55
Explanation:
To find the x intercept, put 0 instead of y,
hence,
0=-0.25x+0.55
-0.25x=-0.55
x=2.2
To find y intercept put 0 instead of x,
so,
y=-0.25*0+0.55
y=0+0.55
y=0.55

3 0

2 years ago

Solve the right triangle. 34.5 Find the length of the side opposite to the given angle. (Round your answer to two decimal places

lapo4ka [179]

Answer:

a) 388.03

b) 148.49

c) π/8

Step-by-step explanation:

Find the diagram attached

Let the opposite side be y

Given

a) Hypotenuse = 420

theta = 3π/8 rad

theta = 3(180)/8

theta = 67.5degrees

Using the SOH CAH TOA identity

sin theta = opposite/hypotenuse

sin 67.5 = y/420

x = 420sin67.5

x = 420(0.9238)

x = 388.03

Hence the length of the side opposite to the given angle is 388.03

b) Hypotenuse = 420

theta = 3π/8 rad

theta = 3(180)/8

theta = 67.5degrees

Using the SOH CAH TOA identity

cos theta = adjacent/hypotenuse

cos 67.5 = x/420

x = 420cos67.5

x = 420(0.3827)

x = 148.49

Hence the length of the side adjacent to the given angle is 148.49

c) The sum of angle in the triangle is π

Let the measure of the unknown angle be z

z + 3π/8 + π/2 = π

z + 3π+4π/8 = π

z + 7π/8 = π

z = π - 7π/8

z = (8π-7π)/8

z = π/8

Hence the measure of the other acute angle is π/8

5 0

2 years ago