1) Simplify the given expressiona) e2 ln 5b) ln ex2c) ln ex2+1d) e2 ln x+cos(2) Solve the given equationa) eln x = 3b) ln e−3x = 3c) ln(2x + 1) = 3d) 2ex+2 = 5e) ln x + ln(x − 1) = ln 2f) 2e−0.2x − 2 = 83) Show that the given functions are inverses of each othera) f (x) = e2x and g (x) = ln √xb) f (x) = ex/2 and g (x) = 2 ln x
Question
- Simplify the given expressiona) e2 ln 5b) ln ex2c) ln ex2+1d) e2 ln x+cos(2) Solve the given equationa) eln x = 3b) ln e−3x = 3c) ln(2x + 1) = 3d) 2ex+2 = 5e) ln x + ln(x − 1) = ln 2f) 2e−0.2x − 2 = 83) Show that the given functions are inverses of each othera) f (x) = e2x and g (x) = ln √xb) f (x) = ex/2 and g (x) = 2 ln x
Solution
Sure, let's solve these problems one by one:
- Simplify the given expressions:
a) e^(2*ln5) = 5^2 = 25
b) ln(e^(x^2)) = x^2
c) ln(e^(x^2+1)) = x^2 + 1
d) e^(2*lnx) + cos(2) = x^2 + cos(2)
- Solve the given equations:
a) e^(lnx) = 3 => x = 3
b) ln(e^(-3x)) = 3 => -3x = 3 => x = -1
c) ln(2x + 1) = 3 => 2x + 1 = e^3 => x = (e^3 - 1)/2
d) 2e^x + 2 = 5 => e^x = 1.5 => x = ln(1.5)
e) ln(x) + ln(x - 1) = ln(2) => ln(x(x - 1)) = ln(2) => x(x - 1) = 2 => x^2 - x - 2 = 0 => x = 1 + sqrt(3), x = 1 - sqrt(3)
f) 2e^(-0.2x) - 2 = 8 => e^(-0.2x) = 5 => -0.2x = ln(5) => x = -5ln(5)
- Show that the given functions are inverses of each other:
a) f(g(x)) = e^(2*ln(sqrt(x))) = x and g(f(x)) = ln(sqrt(e^(2x))) = x, so f and g are inverses of each other.
b) f(g(x)) = e^(2ln(x)/2) = x and g(f(x)) = 2ln(e^(x/2)) = x, so f and g are inverses of each other.
Similar Questions
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Find the exact value of each expression.(a)e−2 ln(5)(b)lnlnee3
Find the inverse of the function y = e5x.On the same diagram sketch the line y = x, and the graphs ofy = e5x and its inverse.(b) Let y = 240 e−7x and set Y = ln y.Find an expression for Y in terms of x and then draw the graph of Yagainst x
(b) Find the exact value of ( )cos p q+ .6. Functions f and g are defined on R by• ( ) 2 5f x x= +• ( ) 2 2g x x x= − .(a) Find an expression for ( )( )f g x .(b) Find an expression for ( )( )g f x .(c) Express ( )( ) ( )( )g f x f g x− in the form ( )2a x b c+ + .123214
l´ımx−→0ln(cos(3x))ln(cos(2x)) =ï 00ò= l´ımx−→0−3 sin(3x)cos(3x)−2 sin(2x)cos(2x)= l´ımx−→0−3 sin(3x) cos(2x)−2 sin(2x) cos(3x)=ï 00ò= 32 l´ımx−→03 cos(3x) cos(2x) − 2 sin(3x) sin(2x)2 cos(2x) cos(3x) − 3 sin(2x) sin(3x) = 32 · 32 = 94
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