Catch up: Powers, Radicals, Exponentials, and Logarithms Flashcards
What is meant by an exponent?
In r^n the number r is called the base number and the number n is called the exponent.
What is the reciprocal of r^2
r^-2
rn ·(r^n)^−1 = r^n ·r^−n = 1,
How do you calculate
2^7 x 2^3 ?
2^7+3 = 2^10 = 1024
How do you calculate
2^-7 x 2^-3 ?
subtract dummy - 1024
r^0 = ?
1
does -r^n = r^-n ?
Obviously not
does r^-n = -1 (r^n)?
no, r^-n = (r^n)^-1
(3^2)^3 = ?
(3^2)^3 = 3^(2*3) = 3^6 = 729
(4 * 12)^2 = ?
(4 * 5)^2 = 4^2 x 5^2 = 16 x 25
(4 / 6) ^ 2 = ?
(4 / 6) ^ 2 = 4^2 / 6^2 = 16 / 36
(4 / 6) ^ -2 = ?
(4 / 6) ^ -2 = (6/4)^2 = 36 / 16
(-3)^-3 = ?
1/27
What is meant by a radical?
Square roots/ roots
sqrt(2) = 2 ^ ?
sqrt(2) = 2 ^ 1/2
(nsqrt(r))^n = ?
(nsqrt(r))^n = (r^1/n)^n = r
-sqrt(r) = ?
-sqrt(r) = sqrt(r) ^ -1
sqrt( r ) ^ 1/3 = ?
sqrt( r ) ^ 1/3 = r ^1/2 * 1/3 = r ^ 1/6
This makes sens because sqrt(r) = r^1/2
sqrt(r x s) = ?
sqrt(r x s) = sqrt(r) x sqrt(s)
If r,s ∈R, r ≥0, s > 0, and n ∈N, then
sqrt(r/s) = ?
sqrt(r) / sqrt(s)
Explain the requirement r ≥0 / r > 0
If r > 0, rα is defined for any real number α ∈ R. We introduced the requirement r ̸= 0 because an negative exponent α expresses a division operation, which is not defined if r = 0.
However, if we limit α to be non-negative (i.e., α ≥0), we can also make sense of rα when r = 0. We introduced the requirement r ≥0 because if r < 0, rα is only defined if α is an odd integer. However, if we limit α to be an odd integer, we can make sense of rα for any r ̸= 0. That is, there is a trade-off between the values that are allowed for r and α.
does nrt(b^m) = nrt(b)^m
Yes you can interchange the order as it is all multiplication and therefore it complies to the commutation property.
Think of it as an exponent:
(b^m)^1/n = b^m/n = (b^1/n)^m
3rt(2^4)= (2^4)^1/3 = 2^4/3 = (2^1/3)^4
does (r + s)n = rn + sn?
No, whereas multiplication and division commute with exponentials, addition and subtraction do not.
If r ∈R,r > 0, then rα is ….
If r ∈R,r ≥0, then rα is …
If r ∈R, then rα is …
If r ∈R,r > 0, then rα is well-defined for all α ∈R,
If r ∈R,r ≥0, then rα is well-defined for all α ∈R,α ≥0,
If r ∈R, then rα is well-defined for all odd α ∈N.
What does it mean to write an expression in standard form?
√x should be rewritten as a√b where a is either an integer or a maximally simplified fraction, and b
is an irreducible root (e.g. √18 = √9 ·√2 = 3√2.)
e.g sqrt(20) = sqrt(4 x 5) = sqrt(4) x sqrt(5) = 2 sqrt(5)
What is the function of logs?
Through exponents and radicals, in the equation s^2 = A we can treat either A or s as unknown. This suggests that, if we know A and s, we can also treat the exponent as the unknown, A = s?. For instance, if A = 8 and s = 2, then 8 = 2r has the solution r = 3, which we denote by log2(8) = r = 3. More generally, we denote the answer to the question A = s? by
logs(A) = r
rewrite 4^3 = 56 as a logarithm
Log4(56) = 3
Name the two special cases that play an important role when doing computations with logarithms
The first special case is when we take r = 1, which gives logb(1) = 0 for any b > 0.
The second special case is if we take r = b, which gives logb(b) = 1. In the section on powers we have already seen that b1 = b for any b > 0.
logb(r1 ·r2) = ?
taking logarithms turns products into sums in the sense that logb(r1 ·r2) = logb(r1) + logb(r2).
logb(r1/r2) = ?
logb(r1) −logb(r2)
logb (r^s) = ?
Logarithms of powers can be expressed as products
logb (r^s) = s logb(r)
by the rule (b^s)^t = b^s·t for powers, is equal to
What is meant by the change-of-base rule?
Finally, logarithms with an arbitrary base b can be expressed in terms of a ratio of
logarithms with a common basis, which is known as the change-of-base rule
logb(r) = log(r) / log(b) .
If r1 = 1, then logb( 1 / r2 ) = ?
−logb(r2)
If r2 ∈R, then logb (r|r2,1| ) = ?
r2 ·logb(r1)
Is log2(2^−3) well defined? Explain
Yes, Firstly, the logarithm logb(r) is only defined for positive r and b. However, if powers appear inside the logarithm, the exponent can very well be negative. For instance, log2(2^−3) = log2(1/8) is well-defined.
log2(4 ·2) = ?
multiplication and division do not commute with taking logarithms; logarithms turn multiplication into summation and division into taking differences:
log2(4 ·2) = log2(8) = log2 (2^3) = 3
log2(4) + log2(2) =
log2 (2^2) + log2(2) = 2 + 1 = 3,
However, if we would multiply the logarithms instead of adding them, we would get log2(4)·log2(2) = 2·1 = 2.
In these notes we will always understand log(r) to mean ____
loge(r)
