Let fx logax use words to describe each of the following tr

Let f(x) = log_a(x), use words to describe each of the following transformations: (a) g(x) = 4 - 5 f(x-2) (b) h(x) = f(3x) - 6 Solve for x if log (x^2-49)-log(x+7) = 1 If log_b x=p and log y=9, find the value of log_xy b in terms of p and g. Solve each of the following for x: (a) 6e^2x = 144; (b) log x + log(x-3) = 1. Given that log_b 2 = 0.3562, log_b 3-0.5646, and log_b 5 = 0.8271, evaluate each of the following: (a) log_b 25; (b) log_b(25/9); (c) log_b Squareroot 30; (d) log_5 Squareroot 5 If y=Ae^kt passes through the points P(0, 2) and Q(4, 3), find the values of A and K. Determine the inverse and the domain of each of the following: (a) f(x) = 85x; (b) h(x) = 4^2x; (c) g(x) = log_5(x-3). Find the time required for an investment to triple in value if it earns 5% annual interest compounded (a) continuous; (b) monthly A certain substance has half-l FE of 60 years. If there is an original amount of 100 grams of the substance, determine value of K to the fourth decimal place. Then determine the time when 20 grams of the substance remains.

Solution

1. f(x)= log_ax

a. g(x)= 4-5f(x-2)

f(x-2)= log_a(x-2)

Therefore g(x)=4-5log_a(x-2)

Since we have -2 inside the function,And in case of minus the graph shifts to right. And the constant that is outside the function is 5 which is greater than 1. In such case the graph is stretched vertically by a factor of 5. An the minus sign that is with 4 reflect it across x axis. ANd in the last because of + 4,the graph shifts to up.

Therefore the transformation is

Shifting 2 units right,Stretching vertically by 5 units,reflection across x axis, Shifting to 4 units up

b. h(x)=f(3x) - 6

h(x)= log_a(3x) - 6

Because of 3 that is multiplication with x,which is greater than 1,the graph compressed horizontally by a factor of 3. And a subtraction of 6 will shift it downwards by a unit of 6.

Therefore the transformation is

compressed horizontally by a factor of 3, shift downwards by 6 units

17. log(x^2-49) - log(x+7)=1

log a- log b=log(a/b)

log(x^2-49/x+7)=1

log ((x+7)(x-7)/(x+7))= 1

log (x-7)= 1

x-7=10^1

x-7=10

x=17