Show that, for all a, b and c, a^log_b (c) = c^log_b (a).

We want to prove:

    alogb(c) = clogb(a).

Recall that we can always write x = eln(x), so xy = (eln(x))y = ey ln(x).

Recall also the change of basis formula for logs:

logb (x) = y  <=>  by = x  <=>  y ln(b) = ln(x)  <=>  y = logb(x) = ln(x) / ln(b).

Putting these two remarks together, we have:

    alogb(c) = elogb(c) ln(a) = e[ln(c) / ln(b)] ln(a) = e[ln(a) / ln(b)] ln(c) = elogb(c) ln(a) = clogb (a).

Q.E.D.

TD
Answered by Tutor69809 D. Maths tutor

5171 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

how do I differentiate?


The curve C has equation y = f(x) where f(x) = (4x + 1) / (x - 2) and x>2. Given that P is a point on C such that f'(x) = -1.


Integrate cos^2A


Integrate 2x^5 + 7x^3 - (3/x^2)


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences