Mathematics High School

How do you prove the change of base formula?

Answers

Answer 1

The change of base formula states that for any logarithm base b and any positive number x: log_b(x) = (log_c(x)) / (log_c(b)), where c is any base.

The change of base formula is a mathematical formula that allows you to convert a logarithm with one base to a logarithm with another base. The formula states that for any logarithm base b and any positive number x: log_b(x) = (log_c(x)) / (log_c(b)), where c is any base.

One way to prove this is to use the properties of logarithms:

log_b(x × y) = log_b(x) + log_b(y)log_b(xⁿ) = n × log_b(x)

Using these properties, we can write log_b(x) as:

log_b(x) = log_b([tex]c^{(log_c(x))}[/tex])

log_b(x) = log_b([tex]c^{(log_c(x))}[/tex]) / log_b(c¹)

log_b(x) = log_c(x) / log_c(b)

Therefore, the change of base formula is proven.

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Related Questions

Simplify the expression -2.2f+0.5f-13-3

Answers

Answer:

-1.7f - 16

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Algebra I

Terms/Coefficients

Step-by-step explanation:

Step 1: Define

-2.2f + 0.5f - 13 - 3

Step 2: Simplify

Combine like terms (f):                                                                                    -1.7f - 13 - 3Combine like terms (Z):                                                                                  -1.7f - 16

Which series of transformations will create similar but not congruent figures? (1 point)
O reflection and translation
O reflection and dilation
O reflection and rotation
O rotation and translation

If you don't know let someone who does answer​

Answers

Answer:

I thik O rotation and translation

Can someone help me please

Answers

Answer:

X=3-(y/2)=8

Step-by-step explanation:

suppose the numbers is X

3 less than half the product of a number is 8

X= 3- (y/2)=8

1/2(8n)-3

Hope that helps

TIMED!!

PLEASE HELP!!

WILL give brainliest

Answers

Answer:

easy third one also plz give heart

Step-by-step explanation:

i just know

Please help, and plz don't be saying this is the right answer if u dont know for sure.

Answers

Answer:

135 degrees

Step-by-step explanation:

(8x-7)+(12x+57)=180

combine like terms

20x+50 = 180

subtract 50 on both sides

20x = 130

divide by 20

x = 130/20

x = 13/2

x = 6.5

cpq = 12x+57

cpq = 78 + 57

cpq = 135

A six-sided Fair number cubis rolled 100 times as part of an experiment. The frequency of the role of the number 3 is 20. W hich statement about rolling a 3 is correct

Answers

Answer:

The Theoretical probability is 1/6.

The experimental probability is 1/5

Step-by-step explanation:

Given

Rolls: 100

Outcome of 3 : 20

Required

Which of the options is true (See attachment for options)

From the attachment, we understand that the question requires that we calculate the experimental and theoretical probability.

The experimental probability (E) is calculated as thus:

[tex]E = \frac{Outcomes}{Total\ Rolls}[/tex]

[tex]E = \frac{20}{100}[/tex]

[tex]E = \frac{1}{5}[/tex]

The theoretical probability (T) is calculated as thus:

A cube has 6 faces, one of which is the face 3.

So, T is:

[tex]T = \frac{n(face\ 3)}{n(Total)}[/tex]

[tex]T = \frac{1}{6}[/tex]

Hence, C is true

what is 3 + 7 -90+180

Answers

Answer:

100

Step-by-step explanation:

3+7=10

-90+180=90

10+90=100

Ten cards numbered 1-10 are placed in a hat.What is the probabiliy of randomly drawing a card with an even number then a card with a number equal to five if the firdt card iis replaced

Answers

Answer:

1/ 20

Step-by-step explanation:

Given :

Cards = {1,2,3,4,5,6,7,8,9,10}

Even number = {2,4,6,8,10}

Probability (drawing an even number) :

Number of even / total. Number of cards

5 / 10 = 1/2

Card with number, 5 :

P(card numbered 5) = 1/10

Since they are independent events :

P(even number) * P(card numbered 5)

1/2 * 1/10 = 1/20

Which statement best describes the domain of the function represented in the graph?

Answers

Given:

The graph of a function.

To find:

The correct statement for the domain of the given function graph.

Solution:

Domain is the set of x-values for which the function is defined or it is the set  of input values.

From the given graph it is clear that it is a linear function with end points (1,2) and (9,14).

So, the given function is defined for [tex]1\leq x\leq 9[/tex]. Here 1 and 9 are including in the set of x-values.

Therefore, the correct option is C, i.e., [tex]1\leq x\leq 9[/tex] or x between 1 and 9, inclusive.

the pythagorean theorem states that a^2 + b^2 = c^2 If A and B represent the length of the legs of a right triangle and C represents the length of the hypotenuse. Find the length of the hypotenuse Is the length of the legs are 9cm and 12cm.​

Answers

Answer:

25 or 50 points is that if 25 or 20 im not

Answer:

The hypotenuse equals 15cm

If 80 = y, then y = 80

Answers

Answeryes

Step-by-step explanation:

FAST!!
Select ALL the correct answers.
With which information can you construct more than one triangle?

the measurements of two angles

the measurements of two angles and the length of the included side

the measurements of all the angles

the lengths of two sides and the measurement of the included angle

Answers

Answer:

a b and d

Step-by-step explanation:

Answer:

the measurements of two angles

the measurements of two angles and the length of the included side

the lengths of two sides and the measurement of the included angle

Step-by-step explanation:

The area of a square field is (49m^2 - 42m + 9) square units. find the dimensions of the field. ( STEP BY STEP EXPLANATION)

Answers

Answer:

(7m−3)(7m−3)

Step-by-step explanation:

Factor 49m2−42m+9

49m2−42m+9

=(7m−3)(7m−3)

I need help with math if you don't know the answer get out of the question and don't answer it and give the wrong answer or some link or even say "I don't know" or anything like that it's not fair to me cause it a waste of my points I will report your if you do that :) have a nice day and stay safe

Answers

Answer:

1 4/6 inches

Step-by-step explanation:

10/6= 1 4/6 inches

Hope it helps!!

Graph the following equation:
Graph f(x)= -2^x-2

Answers

Answer:

Step-by-step explanation:

Find the equation of the linear function represented by the table below in slope-intercept form.

Answers

Answer: The equation is y = 3x + 3

Step-by-step explanation:

Please help
(10 points)
A bill for dinner at a restaurant is $98.87. A 15% tip for the server is added
to the bill. What is the total amount of the bill?
A $100.37
B $109.26
$113.70
D $113.87

Answers

$ 113.70 : 98.87 x .15 = 14.83. ; add that to 98.87

in an assignment and need help with this question

Answers

8 8/3 is 29.3
hope this is correct

The length of a rectangle is 5 centimeters more than the width.
The area of the rectangle is 36 square centimeters.
What is the length?

Answers

Answer:

Length is 4 centimeters

Step-by-step explanation:

4 x (4 + 5)

4 x 9

36

Someone help please

Answers

what do you need help with?

I NEED HELP PLEASE ASAP, It's due soon.

Answers

Answer: y=3/4x+30
(I cannot see the x amounts to see if the units match, if the units are not moving up by 1 on the c axis the slope will be considered wrong)

Why?
Y=mx+b
M=slope
B= y intercept
Slope= rise/run
I moved up from point that directly intersects a point on the graph and moved up 3 and to the right 4 to another point that directly intersects a point.
To find the y intercept I can see that it will directly pass through 30 on the y axis so that is the y intercept

Hope this helps (if I need to explain better or if the x axis doesn’t move by 1s and I need to redo the slope just say something in the comments)

help and ill give extra (60 points)

Answers

Answer:

1) substitution method

2) zero

Step-by-step explanation:

because it has no solution as the variables cancle out eachother.

PLEASE HELP!!

Find the 14th term of the sequence: 2,4,8,16...

Answers

Answer:

16384

Step-by-step explanation:

You have to find 2 to the 14th power (that's the rule)

Hope that helps!

Rewrite the equation 2x + 3y = 6 in slope-intercept form.
3y = -2x + 6
y= -2x+6
y= - 2/3 x+2
y = 2x+6

Answers

2x+3y=6
3y=6-2x
y=2-2/3x
y=-2/3x+2
the 3rd one is correct

please please please help asap

Answers

Answer:

∠ UVX = 126.8°

Step-by-step explanation:

∠ WVS and ∠ UVX are vertical angles and congruent, then

∠ UVX = 126.8°

A Lamborghini is worth $700,732 if the tax of this car is 5% what would be the full price?

Answers

It would be 735,768.6
Because you want to multiply 700,732 by 5% or .05 which would give you 35,036.6 which you then add with 700,732 to get 735,768.6

Solve for x Round to the nearest 100th if needed.
x/13 + 18 = 49

Answers

x
--- + 18 = 49
13


x
--- = 49-18
13

x
--- = 31
13

x = 31×13

x = 403

Answer:

2.4

Step-by-step explanation:

X=49-18/13

X=31/13

X=2.38

I need help on this.. And I will give you a BRANILIST if you the first one with the right answer ​

Answers

Answer:

0.1

Step-by-step explanation:

(9,4) (5,1)    [tex]\frac{y_{2} y_{1} }{x_{2} x_{1} }[/tex], plug in the numbers, [tex]\frac{(1)(4)}{(5)(9)}[/tex]= [tex]\frac{4}{40} or \frac{1}{10}[/tex], to write it as a decimal, you divide the numerator by the denominator, 0.1

Jana has $10. She wants to buy a bottle of nail polish for $9.50. The sales tax is 7%. Does she have enough money?

Answers

Answer:no, she would be 17 cents short

Step-by-step explanation:

Answer:

No.

Step-by-step explanation:

Compute the tax =

9.50 * 7% = 9.5 * .07 or 66.5 cents.

Since she only has $10, she does not have enough money.

Samuel Morse determined that the percentage of t's being used in the English language in the 1800's was 9%. A random sample of 600 letters from a current newspaper contained 48 t's. Test the claim that the proportion of t's has changed in modern times, using the current newspaper data and a significance level of 0.05

Answers

Answer:

There is insufficient evidence to indicate that the population proportion of t’s in modern newspaper is different from 0.09. The result is not statistically significant

Step-by-step explanation:

H0 : p0 = 0.09

H1 : p0 ≠ 0.09

Sample size, n = 600 ; α = 0.05

Decision region

Zstatistic < Zcritical ; reject H0

P = 48 / 600 = 0.08

Zstatistic = (p - p0) ÷ sqrt((p0(1 - p0)) /n)

Zstatistic = (0.08 - 0.09) ÷ sqrt(0.09(0.91))/600)

Zstatistic = - 0.01 /0.0116833

Zstatistic = - 0.856

Zcritical at 0.05, = - 1.96

Zstatistic > Zcritical

-0.856 > - 1.96

Hence, we fail to reject the Null

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