Logarithm!!!

Select all of the following that are true statements:
(a) log(2x) = log(2) + log(x)
(b) log(3x) = 3 log(x)
(c) log(12y) = 2 log(2) + log(3y)
(d) log(5y) = log(20y) – log(4)
(e) log(x) = log(5x) – log(5)
(f) ln(25) = 2ln(5)
(g) ln(1) = log(1)
(h) ln(e) = log(10)
(i) if y = e-3 then ln(3) = ey
(j) if log(x) = -12 then x = 1.00 x 10-12
(k) if ln(x) = -12 then x = 1.00 x 10-12
(l) none of these is correct

2 log 10 power 5 +log 10 power 8-1/2 log 10 power 4 answer

The true statements are:

(a) log(2x) = log(2) + log(x)
(c) log(12y) = 2 log(2) + log(3y)
(f) ln(25) = 2ln(5)
(j) if log(x) = -12 then x = 1.00 x 10-12
(k) if ln(x) = -12 then x = 1.00 x 10-12

Therefore, the correct answer is (a), (c), (f), (j), and (k).

To determine which statements are true, we can apply the rules and properties of logarithms. Let's go through each statement one by one:

(a) log(2x) = log(2) + log(x)
This is a true statement because it follows the rule of logarithms which states that the logarithm of a product is equal to the sum of the logarithms.

(b) log(3x) = 3 log(x)
This is not a true statement. The correct statement should be log(3x) = log(3) + log(x), following the rule of logarithms for multiplication.

(c) log(12y) = 2 log(2) + log(3y)
This is not a true statement. The correct statement should be log(12y) = log(12) + log(y), since the rule of logarithms for multiplication applies to both terms.

(d) log(5y) = log(20y) – log(4)
This is a true statement. The rule of logarithms for division states that the logarithm of a quotient is equal to the difference of the logarithms.

(e) log(x) = log(5x) – log(5)
This is not a true statement. The correct statement should be log(x) = log(5x) - log(5), since the subtraction rule applies.

(f) ln(25) = 2ln(5)
This is a true statement. The property of logarithms states that the natural logarithm of a number raised to a power is equal to the product of the power and the natural logarithm of the number.

(g) ln(1) = log(1)
This is not a true statement. The natural logarithm of 1 is equal to 0, while the logarithm of 1 (in any base) is also equal to 0.

(h) ln(e) = log(10)
This is not a true statement. The natural logarithm of e is equal to 1, while the logarithm of 10 (in any base) is a different value.

(i) if y = e-3 then ln(3) = ey
This is not a true statement. The equation does not represent a direct relationship between ln(3) and ey.

(j) if log(x) = -12 then x = 1.00 × 10-12
This is a true statement. By taking the antilogarithm of both sides, we can determine that x = 1.00 × 10-12.

(k) if ln(x) = -12 then x = 1.00 × 10-12
This is not a true statement. The equation represents a logarithmic relationship, and taking the antilogarithm directly will not give us x = 1.00 × 10-12.

(l) none of these is correct
This is not the correct answer. Some of the statements listed are indeed correct.

So, the true statements are (a), (d), and (j).

type 1 log(xy) = log x + log y

type 2 log (x)^y = y log x

now look at a) type 1, true

b) not type 1, not type 2, false
in fact log 3x = log 3 + log x

c) type 1 and type 2
log (3*2^2* y) = log 3 + 2 log 2 + log y = log 3y + 2 log 2 true

d) type 1
log(20y/4) = log(20y) - log(4)

type 3 base^logbase(x) = x
g) ln(1) = log(1)
e^ln 1 = 1 = 10^log 1
ln 1 = 0 = log 1 true type 3

h) ln(e) = log(10) ?????????
e^lne = e so lne =1
also log 10=1
so true, type 3

i) y = e^-3 do not forget the ^ for exponent
ln y = -3 ln e
ln y = -3 not what you want

j) 10^log x = 10^-12
x = 10^-12 true but remember ^ sign for exponents !

k) e^ln x = x = e^-12 not 10^-12 false