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Prove the change of base formula for logarithms. That is, prove that log_a (x) = log_b (x) / log_b (a).

Firstly, recall the definition of a logarithm: if y = log a (x), then this means that y is the power you have to raise a to, to get x, that is a y = x.Now, we want to introduce a new base, b. Let's take log ...
TH
Answered by Tom H. • Maths tutor
14080 Views

(ii) Prove by induction that, for all positive integers n, f(n) = 3^(3n–2) + 2^(3n+1) is divisible by 19

Let P(n) represent the statement that 'f(n) is divisible by 19'. For the basis step, I prove that P(1) is true: f(1) = 3 3(1)-2 + 2 3(1)+1 = 19. 19 is divisible by 19 so P(1) is true. I now want to prove tha...
DL
Answered by Daniel L. • Maths tutor
10889 Views

How do you find stationary points of an equation, eg. y=x^2+3x+2

Stationary points of an equation are found where the gradient of the tangent at this point equals zero. A diagram can illustrate this. To find them differentiate the given equation (which gives the gradient)...
EC
Answered by Ellie C. • Maths tutor
3866 Views

Find the equation of the tangent to: y = X^2 + 3x + 2 at the point (2,12)

(1) Find the gradient using differentiation (2) If the gradient at (x 1 ,y 1 ) is m,y - y 1 = m(x - x 1 ) (1) We differentiate the given equation:dy/dx = 2x + 3 Then, find the gradient at (2,12). Sub x= 2 in...
SC
Answered by Samuel C. • Maths tutor
4572 Views

Given the equation 0=5x^2+3xy-y^3 find the value of dy/dx at the point (-2,2)

To answer this we will use implicit differentiation with respect to x. So start by differentiating each term. On the left hand side 0 differentiates to 0. On the right hand side 5x 2 differentiates to 10x. B...
HW
Answered by Holly W. • Maths tutor
4189 Views