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Hatshy [7]
2 years ago
15

Using the change-of-base formula, which of the following is equivalent to the

Mathematics
1 answer:
Otrada [13]2 years ago
3 0

Given:

The logarithmic expression is:

\log_718

To find:

The equivalent expression of given expression by using change-of-base formula.

Solution:

The change-of-base formula is:

\log_ba=\dfrac{\log_ca}{\log_cb}

The given expression is:

\log_718

Using the change-of-base formula, we get

\log_718=\dfrac{\log_c18}{\log_c7}

Where, is a constant and c>0.

Let c=10, then

\log_718=\dfrac{\log_{10}18}{\log_{10}7}

Therefore, the correct option is a.

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Find an equation of the tangent to the curve at the given point by both eliminating the parameter and without eliminating the pa
konstantin123 [22]

Answer:

y=2x-8

Step-by-step explanation:

The given parametric equation is;

x=6+ln(t),y=t^2+3

<h3><u>BY ELIMINATING THE PARAMETER</u></h3>

To eliminate the parameter we make t the subject in one equation and put it inside the other.

We make t the subject in x=6+ln(t) because it is easier.

\Rightarrow x-6=ln(t)

\Rightarrow {e}^{x-6}=e^{ln(t)}

\Rightarrow {e}^{x-6}=t

Or

t={e}^{x-6}

We now substitute this into y=t^2+3.

This gives us;

y=(e^{x-6})^2+3.

\Rightarrow y=e^{2(x-6)}+3.

We have now eliminated the parameter.

The equation of the tangent at (6,4) is given by;

y-y_1=m(x-x_1)

where the gradient function is given by;

\frac{dy}{dx}=2e^{2(x-6)}

We substitute x=6 into the gradient function to obtain the gradient.

\Rightarrow m=2e^{2(6-6)}

\Rightarrow m=2e^0

\Rightarrow m=2

The equation of the tangent becomes

y-4=2(x-6)

We simplify to obtain

y=2x-12+4

y=2x-8

<h3><u>WITHOUT ELIMINATING THE PARAMETER</u></h3>

The given parametric equation is;

x=6+ln(t),y=t^2+3

For x=6+ln(t)

\frac{dx}{dt}=\frac{1}{t}

For y=t^2+3

\frac{dy}{dt}=2t

The slope is given by;

\frac{dy}{dx}=\frac{\frac{dy}{dt} }{\frac{dx}{dt} }

\frac{dy}{dx}=\frac{2t }{\frac{1}{t} }

\frac{dy}{dx}=2t^2

At the point, (6,4), we plug in any of the values into the parametric equation and find the corresponding value for t.

Notice that

When x=6, 6=6+\ln(t)

6-6=\ln(t)

0=\ln(t)

e^0=e^\ln(t)

1=t

when y=4, 4=t^2+3

4-3=t^2

1=t^2

t=\pm1

But the slope is the same when we plug in any of these values for t.

\frac{dy}{dx}=2(\pm1)^2=2

The equation of the tangent becomes

y-4=2(x-6)

We simplify to obtain

y=2x-12+4

y=2x-8

7 0
2 years ago
Solve each inequality. Graph the solution on a number line.
Kamila [148]

Step-by-step explanation:

<u>Step 1:  Find the solution for the first one</u>

4x > -36

4x / 4 > -36 / 4

x > -9 ← Solution

<u>Step 2:  Find the solution for the second one</u>

<u />7u\le-49

7y/7\le-49/7

y \le -7 ← Solution

<u>Step 3:  Find the solution for the third one</u>

-5m \ge 15

-5m / -5 \ge 15 / -5

m \le -3 ← Solution

<u>Step 4:  Find the solution for the fourth one</u>

-2p > -3

-2p / -2 > -3 / -2

p < 1.5 ← Solution

<u>Step 5:  Find the solution for the fifth one</u>

45 \le - 10r

45 / -10 \le -10r / -10

-4.5 \ge r ← Solution

<u>Step 6:  Find the solution for the sixth one</u>

<u />-66 \ge -11t

-66 / -11 \ge -11t / -11

6 \le t ← Solution

6 0
2 years ago
Help not much time left its due in 20 mins
marin [14]

Answer:

The slope is 2/3

Step-by-step explanation:

To get from one dot to the next...

You go up 2 units and to the right 3 units

4 0
3 years ago
Read 2 more answers
The altitude of an equilateral triangle is 4.2 meters long. Find the perimeter of the triangle.
earnstyle [38]

Answer:

Step-by-step explanation:

s = √( 4.2*4.2 + (s/2)(s/2) )

s = √( 4.2*4.2 + (s^2/4) )

s^2= 4.2*4.2 + (s^2/4)

s^2 - (s^2/4) = 4.2*4.2

(4s^2/4) - (s^2/4) = 4.2*4.2

(3s^2/4) = 4.2*4.2

(3/4)s^2 = 4.2*4.2

s^2= 4.2*4.2*(4/3)

s = √( 4.2*4.2*(4/3) )

s = 2*4.2 * √( 1/3 )

s = 4.84974226119

answer:

perimeter = 3 * 4.84974226119

perimeter = 14.5492267836 meters

6 0
2 years ago
QUICK I NEED HELP! I WILL MARK BRAINLIEST!
lianna [129]

Answer:

go a head what can i help you with

6 0
2 years ago
Read 2 more answers
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