14 - Exponentials and logs

Question 1

How do you differentiate e^x and e^kx

Answer 1

e^x = e^x

e^kx = ke^kx

the gradient function is the same shape as the graph for e^x

Question 2

What is loga^n = x in power form

Answer 2

a^x = n

(a≠ 1)

Question 3

What is the multiplication of yoga

Answer 3

logA + logB = LogAB

When the base is the same

Question 4

What is the division law

Answer 4

logA - LogB = LogA/B

Where the base is the same

Question 5

What is the power law (logs)

Answer 5

Log(A)^k = kLogA

Question 6

When using the power law how do you solve log(1/x)

Answer 6

This equation looks like it uses the division law, but it does not

=log(x^-1) =-logx

Question 7

What is the solution to loga(a)

Answer 7

Log(a) = 1.

As a^1 = a

Question 8

Why does loga(1) = 0

Answer 8

All values to the power of 0 = 1

Loga(1) = 0

=a^0 = 1

Question 9

What is ln

Answer 9

The natural log (loge)

Question 10

Hows is the graph of ln(x) similar to the graph (e^x)

Answer 10

Ln(x) is a reflection of e^x across the line y=x

Question 11

How would you sove for x, when e^ln(x) = x

Answer 11

Take the natural log of the equation

Ln(e^lnx) = lnx
= ln(e) × (lnx) = lnx
=1lnx = lnx

So
e^lnx = x

This works as loge(e^loge(x) causes the logs and exponentials to cancel out

Question 12

How would you plot the graph of y=ax^n using logs

Answer 12

Logy = loga +n(logx)

The gradient is n
The y-intercept is logA

Question 13

How do you plot the graph of y=ab^x using logs

Answer 13

Logy = xlogb+ loga

Loga is the y intercept
The gradient is logb