Logarythms

Question 1

What does ∀x∈ℝ mean?

Answer 1

‘For all’ values of ‘x’ ‘in’ the ‘real numbers’

Question 2

What’s the modulus symbol?

Answer 2

|x|(means only take the positive value)

Question 3

How do you write log 3 base 2?

Answer 3

log2(3) (2 is small and lower than the other numbers, you could also put a modulus symbol around the 3)

Question 4

How do you work out 2^x=3?

Answer 4

x=log2(3)
x=power/exponent
2=base number
3=the argument (cannot be negative)

Question 5

How do the graphs of y=2^x and y=log2(x) map onto one another?

Answer 5

Reflections in the line y=x

Question 6

What is the equation of the asymptote of y=(3^x)+4

Answer 6

y=4 (y-intercept is 5)

Question 7

Name the logarithmic laws

Answer 7

1) product rule
2) quotient role
3) power rule
(Only apply when the logs involved have the same base!)

Question 8

Describe the product rule for logs

Answer 8

When adding logs with the same base you can multiply the arguments

Question 9

Describe the quotient rule for logs

Answer 9

When subtracting logs with the same base you can divide the arguments

Question 10

Describe the power rule for logs

Answer 10

When you have a number before a log, the answer’s equivalent to the log raised to the power of that number

Question 11

Which logarithmic rule is used first?

Answer 11

The power rule

Question 12

What’s the answer when the base and argument of a log are the same?

Answer 12

1

Question 13

What’s the answer to log b (b)^x and why?

Answer 13

‘x’ because of the power rule it’s equivalent to ‘x’ times log b (b) and a log where the base and argument are the same is equal to 1 so x times 1 = x

Question 14

What is the base of any log automatically assumed to be?

Answer 14

10

Question 15

How do you solve logx(243) = logx(x^5) ?

Answer 15

Equate the arguments of the two equations (since log x is common to both just solve thearguments) 243=x^5 so x=3

Question 16

What is a dummy variable?

Answer 16

Using a letter (usually ‘y’) to represent a more complex expression, allowing you to simplify and solve a difficult equation

Question 17

What can the argument never be?

Answer 17

A negative number or zero

Question 18

What must you say when using dummy variables?

Answer 18

Let ‘y’ equal and then state the more complex expression that you’re replacing e.g. let y=2^x

Question 19

What is ln?

Answer 19

It’s a natural logarithm (behaves the same as logs in calculations same as ‘e’ does) E.g. ln(x) = log e(x)

Question 20

State the value of Euler’s number

Answer 20

2.718

Question 21

State Euler’s identity

Answer 21

(e^iπ) +1 = 0 where i = √-1 and e =2.718

Question 22

What is always the assumed base of ln?

Answer 22

e

Question 23

What is the value of ln 1?

Answer 23

0

Question 24

What is the value of ln e?

Answer 24

1 (because the base is also e)

Question 25

What is the value of ln (e^x)?

Answer 25

x (Because ‘x’ multiplied by ln(e) is the same as ‘x’ multiplied by 1 which is just x)

Question 26

When you multiply numbers with the same bases what happens to their powers?

Answer 26

They’re added! (Common exam mistake)

Question 27

What values of ‘x’ for ln(x) could you not have?

Answer 27

Negatives or zero

Question 28

What is a common mistake I make with logs that’s key to remember in an exam?

Answer 28

**Remember you can equate the arguments when the logs have the same bases!!**

Question 29

What is the log of a fraction with an integer as a base going to equal?

Answer 29

A negative number e.g. log 3 1/27 = -3

Question 30

What is the equation for the line that represents the reflection of the line y = e^x in the line y=x?

Answer 30

y = ln(x)

Question 31

What is another way of writing x = ln(3) / ln(2) ?

Answer 31

Log 3 base 2 ( the numerator becomes the argument and the denominator becomes the base)

Question 32

To satisfy the inequality: (0.3^n) < (0.29^2000) why must the answer of 2056.3 be rounded up to 2057? (Hint you need to solve this to work it out)

Answer 32

Your final line of workings should say n > (2000 x ln(0.29)) / (ln(0.3)) when put into a calculator this means that ‘n’ must be greater than 2056.3 therefore it cannot be rounded down to 2056!!

Question 33

How do you take the logs of both sides in an equation e.g. 2^x=3 ?

Answer 33

ln(2^x) = ln(3) so multiply each side of the equation by ln ( x multiplied by ln(2) = ln(3) x = ln(3) / ln(2) )

Question 34

How do you simplify: (x+3) x ln(5) ?

Answer 34

x(ln(5)) x ln(125) multiply out the brackets to get 2 terms

Question 35

State the gradient function

Answer 35

dy/dx where ‘d’ stands for differentiation

Question 36

State the gradient function of y = 3e^5x

Answer 36

3 x 5 x e^5x = 15e^5x

Question 37

How do you calculate the gradient function?

Answer 37

Coefficient of the base multiplied by coefficient of the power multiplied by the base and powers e.g. for y = 2e^3x + 5 its 2 x 3 x e^3x = 6e^3x

Question 38

What do constants differentiate to?

Answer 38

Zero e.g. y = e^e = 0 when differentiated

Question 39

State the gradient function for y = (e^3x) + 5

Answer 39

1 x 3 x e^3x = 3e^3x **The 5 is completely ignored!!**

Question 40

What axis is y = -e^x reflected in compared to the normal exponential graph?

Answer 40

The x-axis

Question 41

What is the change shown by going from y = e^x to y = e^(x-1)

Answer 41

The graph moves one to the right (and if there’s a +1 after y = e^(x-1) then it would move up one as well)

Question 42

State the factor theorem

Answer 42

If x-a is a factor this is true ⟺ (if and only if) x=a is a root

Question 43

It’s given that (x-3) is a factor of f(x) = x^3-2x^2+ax+1 **find a**

Answer 43

Use the factor theorem. Substitute in 3 for all values of x and solve: a = -10/3

Question 44

Determine whether or not (x-5) is a factor of 4x^4+24x^3-5x^2-150x-125

Answer 44

=4500 which is **NOT equal to zero** therefore (x-5) is not a factor!

Question 45

Find the solution to (e^x) + (3e^-x) = 4

Answer 45

1) multiply by e^x to get rid of the negative power: (e^2x) + 3 = 4e^x 2) rearrange to make it equal to zero: (e^2x) - (4e^x) + 3 = 0 3) factorise to get ((e^x)-3)((e^x)-1) = 0 **(solve like you would a quadratic)** 4) solutions are e^x = 3 or e^x = 1 5) therefore **x = ln(3) or zero**

Question 46

When ln(M) = ln(p) + q ln(t) and q & p are constants identify the gradient and y-intercept

Answer 46

The graph is in the form y = c + mx, the question tells us q & p are constants so they must represent c & m. m = q, c = ln(p)

Question 47

Put **log10(y) = 1/2 log10(x) +1** into the form ax^b

Answer 47

1) log10(y) = log10(x^1/2) +1 2) y = 10^ (log10(x^1/2) +1) **exponentiate using base 10** 3) y = 10^ (log10(x^1/2)) x 10^1(it’s multiply by 10^1 because we’re using power rules not log rules here!) 4) y = x^1/2 x10 5) y = 10x^1/2

Question 48

Reduce y=ab^x to linear form

Answer 48

log10(y) = log10(ab^x) log10(y) = log10(a) + log10(b^x) log10(y) = log10(a) + x log10(b)

Question 49

Rearrange log10(y) = log10(a) + x log10(b) into the form y=mx+c

Answer 49

log10(y) = x log10(b) + log10(a) log10(y) = log10(b) x + log10(a) **the x can be moved behind the log10(b) as multiplication is commutative**

Question 50

Why is this in linear form when plotted: log10(y) = log10(b) x + log10(a) ?

Answer 50

**Because the y value is a log, y= log10(y) so when plotted against x it forms a straight line (the y-intercept is log10(a) and the gradient is log10(b))**

Question 51

Write log10(y) = x + 1/2 in exponential form (ax10^bx)

Answer 51

10^(log10(y)) = 10^(x+1/2) y = 10^(x+1/2) y = 10^x x 10^1/2 this can be commuted to get the answer into the form ax10^bx

Question 52

Given that x=2^p and y=4^q show that log2(yx^3)=3p+2q

Answer 52

Rearrange log2(yx^3): 3log2(x) + log2(y) Substitute in x and y: 3log2(2^p) + log2(4^q) Use the power rule: 3plog2(2) + qlog2(4) = **3p + 2q**

Question 53

Find the exact solution(s) to the equation: 8e^2x -30e^x -27 =0

Answer 53

Factorise: (2e^x -9)(4e^x +3)=0 e^x=9/2 so x=ln(9/2) is the solution because e^x=-3/4 has no solution as the argument of a logarithm can’t be negative

Question 54

A straight line on a graph with axis log T (y) and log l (x) passes through the points: (-0.7,0) and (0.21,0.45). Using this information find a complete equation for the model in the form T=al^b giving the values of ‘a’ and ‘b’ to 3sf

Answer 54

T=al^b is equal to log10(T) = blog10(l) + log10(a) To find log10(a) which is the y-intercept work out the equation of the line y=mx+c gradient is **0.45/0.91 this is the value of b** (change in y/ change in x), insert in the x and y coordinates to get c=9/26 so log10(a)=9/26 thus **a=10^(9/26)=2.22**